# RD Sharma Solutions for Class 11 Chapter 8 - Transformation Formulae Exercise 8.2

In this exercise, we shall discuss problems based on formulae to transform the sum or difference into a product. The solutions prepared by our expert tutors are in an interactive manner to make it easy for the students to understand the concepts. Students can refer to RD Sharma Class 11 Solutions pdf as a major study material to improve their speed in solving problems accurately. Students can download the pdf of RD Sharma Class 11 Maths easily for free from the links given below and can start practising offline for good results.

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1. Express each of the following as the product of sines and cosines:
(i) sin 12x + sin 4x
(ii) sin 5x â€“ sin x
(iii) cos 12x + cos 8x
(iv) cos 12x â€“ cos 4x

(v) sin 2x + cos 4x

Solution:

(i) sin 12x + sin 4x

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 12x + sin 4x = 2 sin (12x + 4x)/2 cos (12x – 4x)/2

= 2 sin 16x/2 cos 8x/2

= 2 sin 8x cos 4x

(ii) sin 5x â€“ sin x

By using the formula,

sin A – sin B = 2 cos (A+B)/2 sin (A-B)/2

sin 5x – sin x = 2 cos (5x + x)/2 sin (5x – x)/2

= 2 cos 6x/2 sin 4x/2

= 2 cos 3x sin 2x

(iii) cos 12x + cos 8x

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

cos 12x + cos 8x = 2 cos (12x + 8x)/2 cos (12x – 8x)/2

= 2 cos 20x/2 cos 4x/2

= 2 cos 10x cos 2x

(iv) cos 12x â€“ cos 4x

By using the formula,

cos A – cos B = -2 sin (A+B)/2 sin (A-B)/2

cos 12x – cos 4x = -2 sin (12x + 4x)/2 sin (12x – 4x)/2

= -2 sin 16x/2 sin 8x/2

= -2 sin 8x sin 4x

(v) sin 2x + cos 4x

sin 2x + cos 4x = sin 2x + sin (90o â€“ 4x)

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 2x + sin (90o â€“ 4x) = 2 sin (2x + 90o – 4x)/2 cos (2x â€“ 90o + 4x)/2

= 2 sin (90o â€“ 2x)/2 cos (6x â€“ 90o)/2

= 2 sin (45Â° – x) cos (3x â€“ 45Â°)

= 2 sin (45Â° – x) cos {-(45Â° – 3x)} (since, {cos (-x) = cos x})

= 2 sin (45Â° – x) cos (45Â° – 3x)

= 2 sin (Ï€/4 – x) cos (Ï€/4 – 3x)

2. Prove that :
(i) sin 38Â° + sin 22Â° = sin 82Â°

(ii) cos 100Â° + cos 20Â° = cos 40Â°

(iii) sin 50Â° + sin 10Â° = cos 20Â°

(iv) sin 23Â° + sin 37Â° = cos 7Â°

(v) sin 105Â° + cos 105Â° = cos 45Â°

(vi) sin 40Â° + sin 20Â° = cos 10Â°

Solution:

(i) sin 38Â° + sin 22Â° = sin 82Â°

Let us consider LHS:

sin 38Â° + sin 22Â°

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 38Â° + sin 22Â° = 2 sin (38o + 22o)/2 cos (38o â€“ 22o)/2

= 2 sin 60o/2 cos 16o/2

= 2 sin 30o cos 8o

= 2 Ã— 1/2 Ã— cos 8o

= cos 8o

= cos (90Â° – 82Â°)

= sin 82Â° (since, {cos (90Â° – A) = sin A})

= RHS

Hence Proved.

(ii) cos 100Â° + cos 20Â° = cos 40Â°

Let us consider LHS:

cos 100Â° + cos 20Â°

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

cos 100Â° + cos 20Â° = 2 cos (100o + 20o)/2 cos (100o â€“ 20o)/2

= 2 cos 120o/2 cos 80o/2

= 2 cos 60o cos 4o

= 2 Ã— 1/2 Ã— cos 40o

= cos 40o

= RHS

Hence Proved.

(iii) sin 50Â° + sin 10Â° = cos 20Â°

Let us consider LHS:

sin 50Â° + sin 10Â°

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 50Â° + sin 10Â° = 2 sin (50o + 10o)/2 cos (50o â€“ 10o)/2

= 2 sin 60o/2 cos 40o/2

= 2 sin 30o cos 20o

= 2 Ã— 1/2 Ã— cos 20o

= cos 20o

= RHS

Hence Proved.

(iv) sin 23Â° + sin 37Â° = cos 7Â°

Let us consider LHS:

sin 23Â° + sin 37Â°

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 23Â° + sin 37Â° = 2 sin (23o + 37o)/2 cos (23o â€“ 37o)/2

= 2 sin 60o/2 cos -14o/2

= 2 sin 30o cos -7o

= 2 Ã— 1/2 Ã— cos -7o

= cos 7o (since, {cos (-A) = cos A})

= RHS

Hence Proved.

(v) sin 105Â° + cos 105Â° = cos 45Â°

Let us consider LHS: sin 105Â° + cos 105Â°

sin 105Â° + cos 105Â° = sin 105o + sin (90o â€“ 105o) [since, {sin (90Â° – A) = cos A}]

= sin 105o + sin (-15o)

= sin 105o â€“ sin 15o [{sin(-A) = – sin A}]

By using the formula,

Sin A â€“ sin B = 2 cos (A+B)/2 sin (A-B)/2

sin 105o â€“ sin 15o = 2 cos (105o + 15o)/2 sin (105o â€“ 15o)/2

= 2 cos 120o/2 sin 90o/2

= 2 cos 60o sin 45o

= 2 Ã— 1/2 Ã— 1/âˆš2

= 1/âˆš2

= cos 45o

= RHS

Hence proved.

(vi) sin 40Â° + sin 20Â° = cos 10Â°

Let us consider LHS:

sin 40Â° + sin 20Â°

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 40Â° + sin 20Â° = 2 sin (40o + 20o)/2 cos (40o â€“ 20o)/2

= 2 sin 60o/2 cos 20o/2

= 2 sin 30o cos 10o

= 2 Ã— 1/2 Ã— cos 10o

= cos 10o

= RHS

Hence Proved.

3. Prove that:

(i) cos 55Â° + cos 65Â° + cos 175Â° = 0

(ii) sin 50Â° â€“ sin 70Â° + sin 10Â° = 0

(iii) cos 80Â° + cos 40Â° â€“ cos 20Â° = 0

(iv) cos 20Â° + cos 100Â° + cos 140Â° = 0

(v) sin 5Ï€/18 â€“ cos 4Ï€/9 = âˆš3 sin Ï€/9

(vi) cos Ï€/12 â€“ sin Ï€/12 = 1/âˆš2

(vii) sin 80Â° â€“ cos 70Â° = cos 50Â°

(viii) sin 51Â° + cos 81Â° = cos 21Â°

Solution:

(i) cos 55Â° + cos 65Â° + cos 175Â° = 0

Let us consider LHS:

cos 55Â° + cos 65Â° + cos 175Â°

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

cos 55Â° + cos 65Â° + cos 175Â° = 2 cos (55o + 65o)/2 cos (55o â€“ 65o) + cos (180o â€“ 5o)

= 2 cos 120o/2 cos (-10o)/2 â€“ cos 5o (since, {cos (180Â° – A) = – cos A})

= 2 cos 60Â° cos (-5Â°) â€“ cos 5Â° (since, {cos (-A) = cos A})

= 2 Ã— 1/2 Ã— cos 5o â€“ cos 5o

= 0

= RHS

Hence Proved.

(ii) sin 50Â° â€“ sin 70Â° + sin 10Â° = 0

Let us consider LHS:

sin 50Â° â€“ sin 70Â° + sin 10Â°

By using the formula,

sin A â€“ sin B = 2 cos (A+B)/2 sin (A-B)/2

sin 50Â° â€“ sin 70Â° + sin 10Â° = 2 cos (50o + 70o)/2 sin (50o â€“ 70o) + sin 10o

= 2 cos 120o/2 sin (-20o)/2 + sin 10o

= 2 cos 60o (- sin 10o) + sin 10o [since,{sin (-A) = -sin (A)}]

= 2 Ã— 1/2 Ã— – sin 10o + sin 10o

= 0

= RHS

Hence proved.

(iii) cos 80Â° + cos 40Â° â€“ cos 20Â° = 0

Let us consider LHS:

cos 80Â° + cos 40Â° â€“ cos 20Â°

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

cos 80Â° + cos 40Â° â€“ cos 20Â° = 2 cos (80o + 40o)/2 cos (80o â€“ 40o) â€“ cos 20o

= 2 cos 120o/2 cos 40o/2 â€“ cos 20o

= 2 cos 60Â° cos 20o â€“ cos 20Â°

= 2 Ã— 1/2 Ã— cos 20o â€“ cos 20o

= 0

= RHS

Hence Proved.

(iv) cos 20Â° + cos 100Â° + cos 140Â° = 0

Let us consider LHS:

cos 20Â° + cos 100Â° + cos 140Â°

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

cos 20Â° + cos 100Â° + cos 140Â° = 2 cos (20o + 100o)/2 cos (20o â€“ 100o) + cos (180o â€“ 40o)

= 2 cos 120o/2 cos (-80o)/2 â€“ cos 40o (since, {cos (180Â° – A) = – cos A})

= 2 cos 60Â° cos (-40Â°) â€“ cos 40Â° (since, {cos (-A) = cos A})

= 2 Ã— 1/2 Ã— cos 40o â€“ cos 40o

= 0

= RHS

Hence Proved.

(v) sin 5Ï€/18 â€“ cos 4Ï€/9 = âˆš3 sin Ï€/9

Let us consider LHS:

sin 5Ï€/18 â€“ cos 4Ï€/9 = sin 5Ï€/18 â€“ sin (Ï€/2 – 4Ï€/9) (since, cos A = sin (90o – A))

= sin 5Ï€/18 â€“ sin (9Ï€ – 8Ï€)/18

= sin 5Ï€/18 â€“ sin Ï€/18

By using the formula,

sin A â€“ sin B = 2 cos (A+B)/2 sin (A-B)/2

= 2 cos (6Ï€/36) sin (4Ï€/36)

= 2 cos Ï€/6 sin Ï€/9

= 2 cos 30o sin Ï€/9

= 2 Ã— âˆš3/2 Ã— sin Ï€/9

= âˆš3 sin Ï€/9

= RHS

Hence proved.

(vi) cos Ï€/12 â€“ sin Ï€/12 = 1/âˆš2

Let us consider LHS:

cos Ï€/12 â€“ sin Ï€/12 = sin (Ï€/2 â€“ Ï€/12) â€“ sin Ï€/12 (since, cos A = sin(90o – A))

= sin (6Ï€ – 5Ï€)/12 â€“ sin Ï€/12

= sin 5Ï€/12 â€“ sin Ï€/12

By using the formula,

sin A â€“ sin B = 2 cos (A+B)/2 sin (A-B)/2

= 2 cos (6Ï€/24) sin (4Ï€/24)

= 2 cos Ï€/4 sin Ï€/6

= 2 cos 45o sin 30o

= 2 Ã— 1/âˆš2 Ã— 1/2

= 1/âˆš2

= RHS

Hence proved.

(vii) sin 80Â° â€“ cos 70Â° = cos 50Â°

sin 80Â° = cos 50Â° + cos 70o

So, now let us consider RHS

cos 50Â° + cos 70o

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

cos 50Â° + cos 70o = 2 cos (50o + 70o)/2 cos (50o â€“ 70o)/2

= 2 cos 120o/2 cos (-20o)/2

= 2 cos 60o cos (-10o)

= 2 Ã— 1/2 Ã— cos 10o (since, cos (-A) = cos A)

= cos 10o

= cos (90Â° – 80Â°)

= sin 80Â° (since, cos (90Â° – A) = sin A)

= LHS

Hence Proved.

(viii) sin 51Â° + cos 81Â° = cos 21Â°

Let us consider LHS:

sin 51Â° + cos 81Â° = sin 51o + sin (90o â€“ 81o)

= sin 51o + sin 9o (since, sin (90Â° – A) = cos A)

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 51o + sin 9o = 2 sin (51o + 9o)/2 cos (51o â€“ 9o)/2

= 2 sin 60o/2 cos 42o/2

= 2 sin 30o cos 21o

= 2 Ã— 1/2 Ã— cos 21o

= cos 21o

= RHS

Hence proved.

4. Prove that:

(i) cos (3Ï€/4 + x) â€“ cos (3Ï€/4 â€“ x) = -âˆš2 sin x

(ii) cos (Ï€/4 + x) + cos (Ï€/4 – x) = âˆš2 cos x

Solution:

(i) cos (3Ï€/4 + x) â€“ cos (3Ï€/4 â€“ x) = -âˆš2 sin x

Let us consider LHS:

cos (3Ï€/4 + x) â€“ cos (3Ï€/4 â€“ x)

By using the formula,

cos A â€“ cos B = -2 sin (A+B)/2 sin (A-B)/2

cos (3Ï€/4 + x) â€“ cos (3Ï€/4 â€“ x) = -2 sin (3Ï€/4 + x + 3Ï€/4 – x)/2 sin (3Ï€/4 + x – 3Ï€/4 + x)/2

= -2 sin (6Ï€/4)/2 sin 2x/2

= -2 sin 6Ï€/8 sin x

= -2 sin 3Ï€/4 sin x

= -2 sin (Ï€ â€“ Ï€/4) sin x

= -2 sin Ï€/4 sin x (since, (Ï€-A) = sin A)

= -2 Ã— 1/âˆš2 Ã— sin x

= -âˆš2 sin x

= RHS

Hence proved.

(ii) cos (Ï€/4 + x) + cos (Ï€/4 – x) = âˆš2 cos x

Let us consider LHS:

cos (Ï€/4 + x) + cos (Ï€/4 – x)

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

cos (Ï€/4 + x) + cos (Ï€/4 – x) = 2 cos (Ï€/4 + x + Ï€/4 – x)/2 cos (Ï€/4 + x – Ï€/4 + x)/2

= 2 cos (2Ï€/4)/2 cos 2x/2

= 2 cos 2Ï€/8 cos x

= 2 sin Ï€/4 cos x

= 2 Ã— 1/âˆš2 Ã— cos x

= âˆš2 cos x

= RHS

Hence proved.

5. Prove that:

(i) sin 65o + cos 65o = âˆš2 cos 20o

(ii) sin 47o + cos 77o = cos 17o

Solution:

(i) sin 65o + cos 65o = âˆš2 cos 20o

Let us consider LHS:

sin 65o + cos 65o = sin 65o + sin (90o â€“ 65o)

= sin 65o + sin 25o

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 65o + sin 25o = 2 sin (65o + 25o)/2 cos (65o â€“ 25o)/2

= 2 sin 90o/2 cos 40o/2

= 2 sin 45o cos 20o

= 2 Ã— 1/âˆš2 Ã— cos 20o

= âˆš2 cos 20o

= RHS

Hence proved.

(ii) sin 47o + cos 77o = cos 17o

Let us consider LHS:

sin 47o + cos 77o = sin 47o + sin (90o â€“ 77o)

= sin 47o + sin 13o

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

sin 47o + sin 13o = 2 sin (47o + 13o)/2 cos (47o â€“ 13o)/2

= 2 sin 60o/2 cos 34o/2

= 2 sin 30o cos 17o

= 2 Ã— 1/2 Ã— cos 17o

= cos 17o

= RHS

Hence proved.

6. Prove that:
(i) cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A

(ii) cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A

(iii) sin A + sin 2A + sin 4A + sin 5A = 4 cos A/2 cos 3A/2 sin 3A

(iv) sin 3A + sin 2A â€“ sin A = 4 sin A cos A/2 cos 3A/2

(v) cos 20o cos 100o + cos 100o cos 140o â€“ cos 140o cos 200o = – 3/4

(vi) sin x/2 sin 7x/2 + sin 3x/2 sin 11x/2 = sin 2x sin 5x

(vii) cos x cos x/2 â€“ cos 3x cos 9x/2 = sin 4x sin 7x/2

Solution:

(i) cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A

Let us consider LHS:

cos 3A + cos 5A + cos 7A + cos 15A

So now,

(cos 5A + cos 3A) + (cos 15A + cos 7A)

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

(cos 5A + cos 3A) + (cos 15A + cos 7A)

= [2 cos (5A+3A)/2 cos (5A-3A)/2] + [2 cos (15A+7A)/2 cos (15A-7A)/2]

= [2 cos 8A/2 cos 2A/2] + [2 cos 22A/2 cos 8A/2]

= [2 cos 4A cos A] + [2 cos 11A cos 4A]

= 2 cos 4A (cos 11A + cos A)

Again by using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

2 cos 4A (cos 11A + cos A) = 2 cos 4A [2 cos (11A+A)/2 cos (11A-A)/2]

= 2 cos 4A [2 cos 12A/2 cos 10A/2]

= 2 cos 4A [2 cos 6A cos 5A]

= 4 cos 4A cos 5A cos 6A

= RHS

Hence proved.

(ii) cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A

Let us consider LHS:

cos A + cos 3A + cos 5A + cos 7A

So now,

(cos 3A + cos A) + (cos 7A + cos 5A)

By using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

(cos 3A + cos A) + (cos 7A + cos 5A)

= [2 cos (3A+A)/2 cos (3A-A)/2] + [2 cos (7A+5A)/2 cos (7A-5A)/2]

= [2 cos 4A/2 cos 2A/2] + [2 cos 12A/2 cos 2A/2]

= [2 cos 2A cos A] + [2 cos 6A cos A]

= 2 cos A (cos 6A + cos 2A)

Again by using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

2 cos A (cos 6A + cos 2A) = 2 cos A [2 cos (6A+2A)/2 cos (6A-2A)/2]

= 2 cos A [2 cos 8A/2 cos 4A/2]

= 2 cos A [2 cos 4A cos 2A]

= 4 cos A cos 2A cos 4A

= RHS

Hence proved.

(iii) sin A + sin 2A + sin 4A + sin 5A = 4 cos A/2 cos 3A/2 sin 3A

Let us consider LHS:

sin A + sin 2A + sin 4A + sin 5A

So now,

(sin 2A + sin A) + (sin 5A + sin 4A)

By using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

(sin 2A + sin A) + (sin 5A + sin 4A) =

= [2 sin (2A+A)/2 cos (2A-A)/2] + [2 sin (5A+4A)/2 cos (5A-4A)/2]

= [2 sin 3A/2 cos A/2] + [2 sin 9A/2 cos A/2]

= 2 cos A/2 (sin 9A/2 + sin 3A/2)

Again by using the formula,

sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

2 cos A/2 (sin 9A/2 + sin 3A/2) = 2 cos A/2 [2 sin (9A/2 + 3A/2)/2 cos (9A/2 â€“ 3A/2)/2]

= 2 cos A/2 [2 sin ((9A+3A)/2)/2 cos ((9A-3A)/2)/2]

= 2 cos A/2 [2 sin 12A/4 cos 6A/4]

= 2 cos A/2 [2 sin 3A cos 3A/2]

= 4 cos A/2 cos 3A/2 sin 3A

= RHS

Hence proved.

(iv) sin 3A + sin 2A â€“ sin A = 4 sin A cos A/2 cos 3A/2

Let us consider LHS:

sin 3A + sin 2A â€“ sin A

So now,

(sin 3A â€“ sin A) + sin 2A

By using the formula,

sin A â€“ sin B = 2 cos (A+B)/2 sin (A-B)/2

(sin 3A â€“ sin A) + sin 2A = 2 cos (3A + A)/2 sin (3A – A)/2 + sin 2A

= 2 cos 4A/2 sin 2A/2 + sin 2A

We know that, sin 2A = 2 sin A cos A

= 2 cos 2A Sin A + 2 sin A cos A

= 2 sin A (cos 2A + cos A)

Again by using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

2 sin A (cos 2A + cos A) = 2 sin A [2 cos (2A+A)/2 cos (2A-A)/2]

= 2 sin A [2 cos 3A/2 cos A/2]

= 4 sin A cos A/2 cos 3A/2

= RHS

Hence proved.

(v) cos 20o cos 100o + cos 100o cos 140o â€“ cos 140o cos 200o = – 3/4

Let us consider LHS:

cos 20o cos 100o + cos 100o cos 140o â€“ cos 140o cos 200o =

We shall multiply and divide by 2 we get,

= 1/2 [2 cos 100o cos 20o + 2 cos 140o cos 100o â€“ 2 cos 200o cos 140o]

We know that 2 cos A cos B = cos (A+B) + cos (A-B)

So,

= 1/2 [cos (100o + 20o) + cos (100o – 20o) + cos (140o + 100o) + cos (140o – 100o) â€“ cos (200o + 140o) â€“ cos (200o – 140o)]]

= 1/2 [cos 120o + cos 80o + cos 240o + cos 40o â€“ cos 340o â€“ cos 60o]

= 1/2 [cos (90o + 30o) + cos 80o + cos (180o + 60o) + cos 40o â€“ cos (360o â€“ 20o) â€“ cos 60o]

We know, cos (180o + A) = – cos A, cos (90o + A) = – sin A, cos (360o – A) = cos A

So,

= 1/2 [- sin 30o + cos 80o â€“ cos 60o + cos 40o â€“ cos 20o â€“ cos 60o]

= 1/2 [- sin 30o + cos 80o + cos 40o â€“ cos 20o â€“ 2 cos 60o]

Again by using the formula,

cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

= 1/2 [- sin 30o + 2 cos (80o+40o)/2 cos (80o-40o)/2 â€“ cos 20o â€“ 2 Ã— 1/2]

= 1/2 [- sin 30o + 2 cos 120o/2 cos 40o/2 â€“ cos 20o – 1]

= 1/2 [- sin 30o + 2 cos 60o cos 20o â€“ cos 20o – 1]

= 1/2 [- 1/2 + 2Ã—1/2Ã—cos 20o â€“ cos 20o – 1]

= 1/2 [-1/2 + cos 20o â€“ cos 20o – 1]

= 1/2 [-1/2 -1]

= 1/2 [-3/2]

= -3/4

= RHS

Hence proved.

(vi) sin x/2 sin 7x/2 + sin 3x/2 sin 11x/2 = sin 2x sin 5x

Let us consider LHS:

sin x/2 sin 7x/2 + sin 3x/2 sin 11x/2 =

We shall multiply and divide by 2 we get,

= 1/2 [2 sin 7x/2 sin x/2 + 2 sin 11x/2 sin 3x/2]

We know that 2 sin A sin B = cos (A-B) – cos (A+B)

So,

= 1/2 [cos (7x/2 â€“ x/2) â€“ cos (7x/2 + x/2) + cos (11x/2 â€“ 3x/2) â€“ cos (11x/2 + 3x/2)]

= 1/2 [cos (7x-x)/2 â€“ cos (7x+x)/2 + cos (11x-3x)/2 â€“ cos (11x+3x)/2]

= 1/2 [cos 6x/2 â€“ cos 8x/2 + cos 8x/2 â€“ cos 14x/2]

= 1/2 [cos 3x â€“ cos 7x]

= – 1/2 [cos 7x â€“ cos 3x]

Again by using the formula,

cos A – cos B = -2 sin (A+B)/2 sin (A-B)/2

= -1/2 [-2 sin (7x+3x)/2 sin (7x-3x)/2]

= -1/2 [-2 sin 10x/2 sin 4x/2]

= -1/2 [-2 sin 5x sin 2x]

= -2/-2 sin 5x sin 2x

= sin 2x sin 5x

= RHS

Hence proved.

(vii) cos x cos x/2 â€“ cos 3x cos 9x/2 = sin 4x sin 7x/2

Let us consider LHS:

cos x cos x/2 â€“ cos 3x cos 9x/2 =

We shall multiply and divide by 2 we get,

= 1/2 [2 cos x cos x/2 â€“ 2 cos 9x/2 cos 3x]

We know that 2 cos A cos B = cos (A+B) + cos (A-B)

So,

= 1/2 [cos (x + x/2) + cos (x â€“ x/2) â€“ cos (9x/2 + 3x) â€“ cos (9x/2 â€“ 3x)]

= 1/2 [cos (2x+x)/2 + cos (2x-x)/2 â€“ cos (9x+6x)/2 â€“ cos (9x-6x)/2]

= 1/2 [cos 3x/2 + cos x/2 â€“ cos 15x/2 â€“ cos 3x/2]

= 1/2 [cos x/2 â€“ cos 15x/2]

= – 1/2 [cos 15x/2 â€“ cos x/2]

Again by using the formula,

cos A – cos B = -2 sin (A+B)/2 sin (A-B)/2

= – 1/2 [-2 sin (15x/2 + x/2)/2 sin (15x/2 â€“ x/2)/2]

= -1/2 [-2 sin (16x/2)/2 sin (14x/2)/2]

= -1/2 [-2 sin 16x/4 sin 7x/2]

= – 1/2 [-2 sin 4x sin 7x/2]

= -2/-2 [sin 4x sin 7x/2]

= sin 4x sin 7x/2

= RHS

Hence proved.

7. Prove that:

Solution:

8. Prove that:

Solution:

= cot 6A

= RHS

Hence proved.

By using the formulas,

sin A â€“ sin B = 2 cos (A+B)/2 sin (A-B)/2

cos A â€“ cos B = -2 sin (A+B)/2 sin (A-B)/2

= RHS

Hence proved.

9. Prove that:

(i) sin Î± + sin Î² + sin Î³ â€“ sin (Î± + Î² + Î³) = 4 sin (Î± + Î²)/2 sin (Î² + Î³)/2 sin (Î± + Î³)/2

(ii) cos (A + B + C) + cos (A â€“ B + C) + cos (A + B â€“ C) + cos (-A + B + C) = 4 cos A cos B cos C

Solution:

(i) sin Î± + sin Î² + sin Î³ â€“ sin (Î± + Î² + Î³) = 4 sin (Î± + Î²)/2 sin (Î² + Î³)/2 sin (Î± + Î³)/2

Let us consider LHS:

sin Î± + sin Î² + sin Î³ â€“ sin (Î± + Î² + Î³)

By using the formulas,

Sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

Sin A â€“ sin B = 2 cos (A+B)/2 sin (A-B)/2

= RHS

Hence proved.

(ii) cos (A + B + C) + cos (A â€“ B + C) + cos (A + B â€“ C) + cos (-A + B + C) = 4 cos A cos B cos C

Let us consider LHS:

cos (A + B + C) + cos (A â€“ B + C) + cos (A + B â€“ C) + cos (-A + B + C)

so,

cos (A + B + C) + cos (A â€“ B + C) + cos (A + B â€“ C) + cos (-A + B + C) =

={cos (A + B + C) + cos (A â€“ B + C)} + {cos (A + B â€“ C) + cos (-A + B + C)}

By using the formula,

Cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

= 4 cos A cos B cos C

= RHS

Hence proved.