NCERT Solutions for Class 10 Maths Chapter 8 - Introduction to Trigonometry Exercise 8.4

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Access other exercise solutions of Class 10 Maths Chapter 8 – Introduction to Trigonometry

Exercise 8.1 Solutions – 11 Questions (7 short answers, 3 long answers, 1 short answer with reasoning)

Exercise 8.2 Solutions – 4 Questions ( 1 short answer, 2 long answers, 1 MCQ)

Exercise 8.3 Solutions – 7 Questions (5 short answers, 2 long answers)

Access answers of Maths NCERT Class 10 Chapter 8 – Introduction to Trigonometry Exercise 8.4

1. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Solution:

To convert the given trigonometric ratios in terms of the cot functions, use trigonometric formulas.

We know that,

cosec²A – cot²A = 1

cosec²A = 1 + cot²A

Since the cosec function is the inverse of the sin function, it is written as

1/sin²A = 1 + cot²A

Now, rearrange the terms; it becomes

sin²A = 1/(1+cot²A)

Now, take square roots on both sides; we get

sin A = ±1/(√(1+cot²A)

The above equation defines the sin function in terms of the cot function

Now, to express the sec function in terms of the cot function, use the formula

sin²A = 1/ (1+cot²A)

Now, represent the sin function as the cos function

1 – cos²A = 1/ (1+cot²A)

Rearrange the terms,

cos²A = 1 – 1/(1+cot²A)

⇒cos²A = (1-1+cot²A)/(1+cot²A)

Since the sec function is the inverse of the cos function,

⇒ 1/sec²A = cot²A/(1+cot²A)

Take the reciprocal and square roots on both sides, and we get

⇒ sec A = ±√ (1+cot²A)/cotA

Now, to express the tan function in terms of the cot function

tan A = sin A/cos A and cot A = cos A/sin A

Since the cot function is the inverse of the tan function, it is rewritten as

tan A = 1/cot A

2. Write all the other trigonometric ratios of ∠A in terms of sec A.

Solution:

Cos A function in terms of sec A:

sec A = 1/cos A

⇒ cos A = 1/sec A

sec A function in terms of sec A:

cos²A + sin²A = 1

Rearrange the terms.

sin²A = 1 – cos²A

sin²A = 1 – (1/sec²A)

sin²A = (sec²A-1)/sec²A

sin A = ± √(sec²A-1)/sec A

cosec A function in terms of sec A:

sin A = 1/cosec A

⇒cosec A = 1/sin A

cosec A = ± sec A/√(sec²A-1)

Now, tan A function in terms of sec A:

sec²A – tan²A = 1

Rearrange the terms.

⇒ tan²A = sec²A – 1

tan A = √(sec²A – 1)

cot A function in terms of sec A:

tan A = 1/cot A

⇒ cot A = 1/tan A

cot A = ±1/√(sec²A – 1)

3. Evaluate:

(i) (sin²63° + sin²27°)/(cos²17° + cos²73°)
(ii)  sin 25° cos 65° + cos 25° sin 65°

Solution:

(i) (sin²63° + sin²27°)/(cos²17° + cos²73°)

To simplify this, convert some of the sin functions into cos functions and cos functions into sin functions, and they become,

= [sin²(90°-27°) + sin²27°] / [cos²(90°-73°) + cos²73°)]

= (cos²27° + sin²27°)/(sin²27° + cos²73°)

= 1/1 =1                       (since sin²A + cos²A = 1)

Therefore, (sin²63° + sin²27°)/(cos²17° + cos²73°) = 1

(ii) sin 25° cos 65° + cos 25° sin 65°

To simplify this, convert some of the sin functions into cos functions and cos functions into sin functions, and they become,

= sin(90°-25°) cos 65° + cos (90°-65°) sin 65°

= cos 65° cos 65° + sin 65° sin 65°

= cos²65° + sin²65° = 1 (since sin²A + cos²A = 1)

Therefore, sin 25° cos 65° + cos 25° sin 65° = 1

4. Choose the correct option. Justify your choice.
(i) 9 sec²A – 9 tan²A =
(A) 1                 (B) 9              (C) 8                (D) 0
(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ)
(A) 0                 (B) 1              (C) 2                (D) – 1
(iii) (sec A + tan A) (1 – sin A) =
(A) sec A           (B) sin A        (C) cosec A      (D) cos A

(iv) 1+tan²A/1+cot²A = 

      (A) sec² A                 (B) -1              (C) cot²A                (D) tan²A

Solution:

(i) (B) is correct.

Justification:

Take 9 outside, and it becomes

9 sec²A – 9 tan²A

= 9 (sec²A – tan²A)

= 9×1 = 9             (∵ sec2 A – tan2 A = 1)

Therefore, 9 sec²A – 9 tan²A = 9

(ii) (C) is correct

Justification:

(1 + tan θ + sec θ) (1 + cot θ – cosec θ)

We know that tan θ = sin θ/cos θ

sec θ = 1/ cos θ

cot θ = cos θ/sin θ

cosec θ = 1/sin θ

Now, substitute the above values in the given problem, and we get

= (1 + sin θ/cos θ + 1/ cos θ) (1 + cos θ/sin θ – 1/sin θ)

Simplify the above equation.

= (cos θ +sin θ+1)/cos θ × (sin θ+cos θ-1)/sin θ

= (cos θ+sin θ)²-1²/(cos θ sin θ)

= (cos²θ + sin²θ + 2cos θ sin θ -1)/(cos θ sin θ)

= (1+ 2cos θ sin θ -1)/(cos θ sin θ) (Since cos²θ + sin²θ = 1)

= (2cos θ sin θ)/(cos θ sin θ) = 2

Therefore, (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =2

(iii) (D) is correct.

Justification:

We know that,

Sec A= 1/cos A

Tan A = sin A/cos A

Now, substitute the above values in the given problem, and we get

(secA + tanA) (1 – sinA)

= (1/cos A + sin A/cos A) (1 – sinA)

= (1+sin A/cos A) (1 – sinA)

= (1 – sin²A)/cos A

= cos²A/cos A = cos A

Therefore, (secA + tanA) (1 – sinA) = cos A

(iv) (D) is correct.

Justification:

We know that,

tan²A =1/cot²A

Now, substitute this in the given problem, and we get

1+tan²A/1+cot²A

= (1+1/cot²A)/1+cot²A

= (cot²A+1/cot²A)×(1/1+cot²A)

= 1/cot²A = tan²A

So, 1+tan²A/1+cot²A = tan²A

5. Prove the following identities, where the angles involved are acute angles for which the
expressions are defined.

(i) (cosec θ – cot θ)² = (1-cos θ)/(1+cos θ)

(ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A

(iii) tan θ/(1-cot θ) + cot θ/(1-tan θ) = 1 + sec θ cosec θ

     [Hint: Write the expression in terms of sin θ and cos θ]

(iv) (1 + sec A)/sec A = sin²A/(1-cos A)  

     [Hint: Simplify L.H.S. and R.H.S. separately]

(v) ( cos A–sin A+1)/( cos A +sin A–1) = cosec A + cot A, using the identity cosec²A = 1+cot²A.

(vii) (sin θ – 2sin³θ)/(2cos³θ-cos θ) = tan θ
(viii) (sin A + cosec A)² + (cos A + sec A)² = 7+tan²A+cot²A
(ix) (cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA)
[Hint: Simplify L.H.S. and R.H.S. separately] (x) (1+tan²A/1+cot²A) = (1-tan A/1-cot A)² = tan²A

Solution:

(i) (cosec θ – cot θ)² = (1-cos θ)/(1+cos θ)

To prove this, first, take the Left Hand Side (L.H.S.) of the given equation to prove the Right Hand Side (R.H.S.)

L.H.S. = (cosec θ – cot θ)²

The above equation is in the form of (a-b)², and expand it

Since (a-b)² = a² + b² – 2ab

Here, a = cosec θ and b = cot θ

= (cosec²θ + cot²θ – 2cosec θ cot θ)

Now, apply the corresponding inverse functions and equivalent ratios to simplify

= (1/sin²θ + cos²θ/sin²θ – 2cos θ/sin²θ)

= (1 + cos²θ – 2cos θ)/(1 – cos²θ)

= (1-cos θ)²/(1 – cosθ)(1+cos θ)

= (1-cos θ)/(1+cos θ) = R.H.S.

Therefore, (cosec θ – cot θ)² = (1-cos θ)/(1+cos θ)

Hence proved.

(ii)  (cos A/(1+sin A)) + ((1+sin A)/cos A) = 2 sec A

Now, take the L.H.S. of the given equation.

L.H.S. = (cos A/(1+sin A)) + ((1+sin A)/cos A)

= [cos²A + (1+sin A)²]/(1+sin A)cos A

= (cos²A + sin²A + 1 + 2sin A)/(1+sin A) cos A

Since cos²A + sin²A = 1, we can write it as

= (1 + 1 + 2sin A)/(1+sin A) cos A

= (2+ 2sin A)/(1+sin A)cos A

= 2(1+sin A)/(1+sin A)cos A

= 2/cos A = 2 sec A = R.H.S.

L.H.S. = R.H.S.

(cos A/(1+sin A)) + ((1+sin A)/cos A) = 2 sec A

Hence proved.

(iii) tan θ/(1-cot θ) + cot θ/(1-tan θ) = 1 + sec θ cosec θ

L.H.S. = tan θ/(1-cot θ) + cot θ/(1-tan θ)

We know that tan θ =sin θ/cos θ

cot θ = cos θ/sin θ

Now, substitute it in the given equation to convert it into a simplified form.

= [(sin θ/cos θ)/1-(cos θ/sin θ)] + [(cos θ/sin θ)/1-(sin θ/cos θ)]

= [(sin θ/cos θ)/(sin θ-cos θ)/sin θ] + [(cos θ/sin θ)/(cos θ-sin θ)/cos θ]

= sin²θ/[cos θ(sin θ-cos θ)] + cos²θ/[sin θ(cos θ-sin θ)]

= sin²θ/[cos θ(sin θ-cos θ)] – cos²θ/[sin θ(sin θ-cos θ)]

= 1/(sin θ-cos θ) [(sin²θ/cos θ) – (cos²θ/sin θ)]

= 1/(sin θ-cos θ) × [(sin³θ – cos³θ)/sin θ cos θ]

= [(sin θ-cos θ)(sin²θ+cos²θ+sin θ cos θ)]/[(sin θ-cos θ)sin θ cos θ]

= (1 + sin θ cos θ)/sin θ cos θ

= 1/sin θ cos θ + 1

= 1 + sec θ cosec θ = R.H.S.

Therefore, L.H.S. = R.H.S.

Hence proved.

(iv)  (1 + sec A)/sec A = sin²A/(1-cos A)

First, find the simplified form of L.H.S.

L.H.S. = (1 + sec A)/sec A

Since the secant function is the inverse function of the cos function, it is written as

= (1 + 1/cos A)/1/cos A

= (cos A + 1)/cos A/1/cos A

Therefore, (1 + sec A)/sec A = cos A + 1

R.H.S. = sin²A/(1-cos A)

We know that sin²A = (1 – cos²A), we get

= (1 – cos²A)/(1-cos A)

= (1-cos A)(1+cos A)/(1-cos A)

Therefore, sin²A/(1-cos A)= cos A + 1

L.H.S. = R.H.S.

Hence proved.

(v) (cos A–sin A+1)/(cos A+sin A–1) = cosec A + cot A, using the identity cosec²A = 1+cot²A.

With the help of the identity function, cosec²A = 1+cot²A, let us prove the above equation.

L.H.S. = (cos A–sin A+1)/(cos A+sin A–1)

Divide the numerator and denominator by sin A, and we get

= (cos A–sin A+1)/sin A/(cos A+sin A–1)/sin A

We know that cos A/sin A = cot A and 1/sin A = cosec A

= (cot A – 1 + cosec A)/(cot A+ 1 – cosec A)

= (cot A – cosec²A + cot²A + cosec A)/(cot A+ 1 – cosec A) (using cosec²A – cot²A = 1

= [(cot A + cosec A) – (cosec²A – cot²A)]/(cot A+ 1 – cosec A)

= [(cot A + cosec A) – (cosec A + cot A)(cosec A – cot A)]/(1 – cosec A + cot A)

=  (cot A + cosec A)(1 – cosec A + cot A)/(1 – cosec A + cot A)

=  cot A + cosec A = R.H.S.

Therefore, (cos A–sin A+1)/(cos A+sin A–1) = cosec A + cot A

Hence proved.

First, divide the numerator and denominator of L.H.S. by cos A,

We know that 1/cos A = sec A and sin A/ cos A = tan A, and it becomes,

= √(sec A+ tan A)/(sec A-tan A)

Now using rationalisation, we get

= (sec A + tan A)/1

= sec A + tan A = R.H.S.

Hence proved.

(vii) (sin θ – 2sin³θ)/(2cos³θ-cos θ) = tan θ

L.H.S. = (sin θ – 2sin³θ)/(2cos³θ – cos θ)

Take sin θ as in numerator and cos θ in the denominator as outside; it becomes

= [sin θ(1 – 2sin²θ)]/[cos θ(2cos²θ- 1)]

We know that sin²θ = 1-cos²θ

= sin θ[1 – 2(1-cos²θ)]/[cos θ(2cos²θ -1)]

= [sin θ(2cos²θ -1)]/[cos θ(2cos²θ -1)]

= tan θ = R.H.S.

Hence proved.

(viii) (sin A + cosec A)² + (cos A + sec A)² = 7+tan²A+cot²A

L.H.S. = (sin A + cosec A)² + (cos A + sec A)²

It is of the form (a+b)², expand it

(a+b)² =a² + b² +2ab

               = (sin²A + cosec²A + 2 sin A cosec A) + (cos²A + sec²A + 2 cos A sec A)

= (sin²A + cos²A) + 2 sin A(1/sin A) + 2 cos A(1/cos A) + 1 + tan²A + 1 + cot²A

= 1 + 2 + 2 + 2 + tan²A + cot²A

= 7+tan²A+cot²A = R.H.S.

Therefore, (sin A + cosec A)² + (cos A + sec A)² = 7+tan²A+cot²A

Hence proved.

(ix) (cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA)

First, find the simplified form of L.H.S.

L.H.S. = (cosec A – sin A)(sec A – cos A)

Now, substitute the inverse and equivalent trigonometric ratio forms.

= (1/sin A – sin A)(1/cos A – cos A)

= [(1-sin²A)/sin A][(1-cos²A)/cos A]

= (cos²A/sin A)×(sin²A/cos A)

= cos A sin A

Now, simplify the R.H.S.

R.H.S. = 1/(tan A+cotA)

= 1/(sin A/cos A +cos A/sin A)

= 1/[(sin²A+cos²A)/sin A cos A]

= cos A sin A

L.H.S. = R.H.S.

(cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA)

Hence proved.

(x)  (1+tan²A/1+cot²A) = (1-tan A/1-cot A)² = tan²A

L.H.S. = (1+tan²A/1+cot²A)

Since the cot function is the inverse of the tan function,

= (1+tan²A/1+1/tan²A)

= 1+tan²A/[(1+tan²A)/tan²A]

Now cancel the 1+tan²A terms, and we get

= tan²A

(1+tan²A/1+cot²A) = tan²A

Similarly,

(1-tan A/1-cot A)² = tan²A

Hence proved.

Exercise 8.4 of NCERT Solutions for Class 10 Maths recalls the concept of trigonometric identities. In this exercise, a trigonometric identity is defined as an equation which involves the trigonometric ratios of an angle when it is true for all the values of the angle involved in a right-angle triangle. With the help of trigonometric identities, the expression of one trigonometric ratio is written in terms of other trigonometric ratios. By finding one trigonometric identity of the cos and sin functions, the trigonometric identities of other functions are easily found since the functions are interrelated to each other, and the proofs are given elaborately.

In this chapter, trigonometric ratios are expressed in terms of other ratios, proving the trigonometric expressions and justifications with respect to the trigonometric identities given. Also, refer to NCERT solutions for Class 10 maths chapter 8 to solve more problems to score high in the board examinations.