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

Exercise 8.2 Solutions â€“ 4 Questions ( 1 short answer, 2 long answers, 1 MCQ)

Exercise 8.3 Solutions â€“ 7 Questions (5 short answers, 2 long answers)

### Access Answers of Maths NCERT Class 10 Chapter 8 â€“ 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 cot functions, use trigonometric formulas

We know that,

cosec^{2}A^{Â }â€“ cot^{2}A = 1

cosec^{2}A = 1Â + cot^{2}A

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

1/sin^{2}A = 1Â + cot^{2}A

Now, rearrange the terms, it becomes

sin^{2}A = 1/(1+cot^{2}A)

Now, take square roots on both sides, we get

sin A = Â±1/(âˆš(1+cot^{2}A)

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

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

sin^{2}A = 1/ (1+cot^{2}A)

Now, represent the sin function as cos function

1 â€“ cos^{2}A = 1/ (1+cot^{2}A)

Rearrange the terms,

cos^{2}A = 1 â€“ 1/(1+cot^{2}A)

â‡’cos^{2}A =Â (1-1+cot^{2}A)/(1+cot^{2}A)

Since sec function is the inverse of cos function,

â‡’ 1/sec^{2}A = cot^{2}A/(1+cot^{2}A)

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

â‡’ sec A = Â±âˆš (1+cot^{2}A)/cotA

Now, to express tan function in terms of cot function

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

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

tan A = 1/cot A

**2.Â Write all the other trigonometric ratios of âˆ A in terms of sec A.**

Solution:

Cos A function in terms of sec A:

sec A = 1/cos A

â‡’ cos A = 1/sec A

sec A function in terms of sec A:

cos^{2}AÂ + sin^{2}A = 1

Rearrange the terms

sin^{2}A = 1 â€“ cos^{2}A

sin^{2}A = 1 â€“ (1/sec^{2}A)

sin^{2}A = (sec^{2}A-1)/sec^{2}A

sin A = Â± âˆš(sec^{2}A-1)/sec A

cosec A function in terms of sec A:

sin A = 1/cosec A

â‡’cosec A = 1/sin A

cosec A = Â± sec A/âˆš(sec^{2}A-1)

Now, tan A function in terms of sec A:

sec^{2}A â€“ tan^{2}A = 1

Rearrange the terms

â‡’ tan^{2}A = sec^{2}AÂ + 1

tan A = âˆš(sec^{2}AÂ + 1)

cot A function in terms of sec A:

tan A = 1/cot A

â‡’ cot A = 1/tan A

cot A = Â±1/âˆš(sec^{2}AÂ + 1)

**3.Â Evaluate:**

**(i) (sin ^{2}63Â° + sin^{2}27Â°)/(cos^{2}17Â° + cos^{2}73Â°)**

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

Solution:

(i) (sin^{2}63Â° + sin^{2}27Â°)/(cos^{2}17Â° + cos^{2}73Â°)

To simplify this, convert some of the sin functions into cos functions and cos function into sin function and it becomes,

= [sin^{2}(90Â°-27Â°) + sin^{2}27Â°] / [cos^{2}(90Â°-73Â°) + cos^{2}73Â°)]

= (cos^{2}27Â°^{Â }+ sin^{2}27Â°)/(sin^{2}27Â°Â + cos^{2}73Â°)

= 1/1 =1 Â Â Â Â Â Â Â Â Â Â Â (since sin^{2}AÂ + cos^{2}A = 1)

Therefore, (sin^{2}63Â° + sin^{2}27Â°)/(cos^{2}17Â° + cos^{2}73Â°) = 1

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

To simplify this, convert some of the sin functions into cos functions and cos function into sin function and it becomes,

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

= cos 65Â° cos 65Â°Â + sin 65Â° sin 65Â°

= cos^{2}65Â°^{Â }+ sin^{2}65Â° = 1 (since sin^{2}AÂ + cos^{2}A = 1)

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

**4. Choose the correct option. Justify your choice.
(i) 9 sec ^{2}A â€“ 9 tan^{2}A =
(A) 1 Â Â Â Â Â Â Â Â Â (B) 9 Â Â Â Â Â Â Â (C) 8 Â Â Â Â Â Â Â Â (D) 0
(ii) (1 + tan Î¸ + sec Î¸) (1 + cot Î¸ â€“ cosec Î¸)
(A) 0 Â Â Â Â Â Â Â Â (B) 1 Â Â Â Â Â Â Â (C) 2 Â Â Â Â Â Â Â Â (D) â€“ 1
(iii) (sec A + tan A) (1 â€“ sin A) =
(A) sec A Â Â Â Â Â Â (B) sin A Â Â Â Â (C) cosec A Â Â Â (D) cos A**

**(iv) 1+tan ^{2}A/1+cot^{2}A =Â **

**Â Â Â (A)Â sec ^{2 }AÂ Â Â Â Â Â Â Â Â (B) -1 Â Â Â Â Â Â Â (C)Â cot^{2}AÂ Â Â Â Â Â Â Â Â (D) tan^{2}A**

Solution:

(i) (B) is correct.

Justification:

Take 9 outside, and it becomes

9Â sec^{2}AÂ â€“ 9 tan^{2}A

= 9 (sec^{2}AÂ â€“ tan^{2}A)

=Â 9Ã—1 =Â 9 Â Â Â Â Â Â Â (âˆµÂ sec2 A â€“ tan2 A = 1)

Therefore, 9Â sec^{2}AÂ â€“ 9 tan^{2}A = 9

(ii) (C) is correct

Justification:

(1 + tan Î¸ + sec Î¸) (1 + cot Î¸ â€“ cosec Î¸)

We know that, tan Î¸ = sin Î¸/cos Î¸

sec Î¸ = 1/ cos Î¸

cot Î¸ = cos Î¸/sin Î¸

cosec Î¸ = 1/sin Î¸

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

= (1Â + sin Î¸/cos Î¸Â + 1/ cos Î¸) (1Â + cos Î¸/sin Î¸ â€“ 1/sin Î¸)

Simplify the above equation,

= (cos Î¸ +sin Î¸+1)/cos Î¸Â Ã— (sin Î¸+cos Î¸-1)/sin Î¸

= (cos Î¸+sin Î¸)^{2}-1^{2}/(cos Î¸ sin Î¸)

= (cos^{2}Î¸ + sin^{2}Î¸ + 2cos Î¸ sin Î¸ -1)/(cos Î¸ sin Î¸)

= (1+ 2cos Î¸ sin Î¸ -1)/(cos Î¸ sin Î¸) (Since cos^{2}Î¸ + sin^{2}Î¸ = 1)

= (2cos Î¸ sin Î¸)/(cos Î¸ sin Î¸) = 2

Therefore, (1 + tan Î¸ + sec Î¸) (1 + cot Î¸ â€“ cosec Î¸) =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, we get

(secA + tanA) (1 â€“ sinA)

= (1/cos AÂ + sin A/cos A) (1 â€“ sinA)

= (1+sin A/cos A) (1 â€“ sinA)

= (1 â€“ sin^{2}A)/cos A

= cos^{2}A/cos A = cos A

Therefore, (secA + tanA) (1 â€“ sinA) = cos A

(iv)Â (D) is correct.

Justification:

We know that,

tan^{2}A =1/cot^{2}A

Now, substitute this in the given problem, we get

1+tan^{2}A/1+cot^{2}A

= (1+1/cot^{2}A)/1+cot^{2}A

= (cot^{2}A+1/cot^{2}A)Ã—(1/1+cot^{2}A)

=Â 1/cot^{2}A = tan^{2}A

So, 1+tan^{2}A/1+cot^{2}A = tan^{2}A

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

**(i) (cosec Î¸ â€“ cot Î¸) ^{2Â }= (1-cos Î¸)/(1+cos Î¸)**

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

**(iii) tan Î¸/(1-cot Î¸)Â + cot Î¸/(1-tan Î¸) = 1Â + sec Î¸ cosec Î¸**

**Â Â Â [Hint : Write the expression in terms of sin Î¸ and cos Î¸]**

**(iv) (1Â + sec A)/sec A = sin ^{2}A/(1-cos A) Â **

**Â Â Â [Hint : Simplify LHS and RHS separately]**

**(v) ( cos Aâ€“sin A+1)/( cos A +sin Aâ€“1) =Â cosec A + cot A, using the identity cosec ^{2}A = 1+cot^{2}A.**

** (vii) (sin Î¸ â€“ 2sin ^{3}Î¸)/(2cos^{3}Î¸-cos Î¸) = tan Î¸**

(viii) (sin AÂ + cosec A)^{2Â }+ (cos A + sec A)^{2}Â =Â 7+tan^{2}A+cot^{2}A

(ix)Â (cosec A â€“ sin A)(sec A â€“ cos A) = 1/(tan A+cotA)

[Hint : Simplify LHS and RHS separately]

(x) (1+tan^{2}A/1+cot^{2}A) = (1-tan A/1-cot A)^{2}Â =^{Â }tan^{2}A

Solution:

(i) (cosec Î¸ â€“ cot Î¸)^{2Â }= (1-cos Î¸)/(1+cos Î¸)

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 Î¸ â€“ cot Î¸)^{2}

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

Since (a-b)^{2} = a^{2} + b^{2} â€“ 2ab

Here a = cosec Î¸ and b = cot Î¸

= (cosec^{2}Î¸ +Â cot^{2}Î¸ â€“ 2cosec Î¸ cot Î¸)

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

= (1/sin^{2}Î¸Â + cos^{2}Î¸/sin^{2}Î¸ â€“ 2cos Î¸/sin^{2}Î¸)

= (1 + cos^{2}Î¸ â€“ 2cos Î¸)/(1 â€“ cos^{2}Î¸)

= (1-cos Î¸)^{2}/(1 â€“ cosÎ¸)(1+cos Î¸)

= (1-cos Î¸)/(1+cos Î¸) = R.H.S.

Therefore, (cosec Î¸ â€“ cot Î¸)^{2Â }= (1-cos Î¸)/(1+cos Î¸)

Hence proved.

(ii) Â (cos A/(1+sin A))Â + ((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))Â + ((1+sin A)/cos A)

= [cos^{2}AÂ + (1+sin A)^{2}]/(1+sin A)cos A

= (cos^{2}A + sin^{2}A + 1Â + 2sin A)/(1+sin A) cos A

Since cos^{2}A + sin^{2}A = 1, we can write it as

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

Hence proved.

(iii) tan Î¸/(1-cot Î¸)Â + cot Î¸/(1-tan Î¸) = 1Â + sec Î¸ cosec Î¸

L.H.S. = tan Î¸/(1-cot Î¸)Â + cot Î¸/(1-tan Î¸)

We know that tan Î¸ =sin Î¸/cos Î¸

cot Î¸ = cos Î¸/sin Î¸

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

= [(sin Î¸/cos Î¸)/1-(cos Î¸/sin Î¸)]Â + [(cos Î¸/sin Î¸)/1-(sinÂ Î¸/cos Î¸)]

= [(sin Î¸/cos Î¸)/(sin Î¸-cos Î¸)/sin Î¸]Â + [(cos Î¸/sin Î¸)/(cos Î¸-sinÂ Î¸)/cos Î¸]

= sin^{2}Î¸/[cos Î¸(sin Î¸-cos Î¸)] + cos^{2}Î¸/[sin Î¸(cos Î¸-sin Î¸)]

= sin^{2}Î¸/[cos Î¸(sin Î¸-cos Î¸)] â€“ cos^{2}Î¸/[sin Î¸(sin Î¸-cos Î¸)]

= 1/(sin Î¸-cos Î¸) [(sin^{2}Î¸/cos Î¸) â€“ (cos^{2}Î¸/sin Î¸)]

= 1/(sin Î¸-cos Î¸)Â Ã— [(sin^{3}Î¸ â€“ cos^{3}Î¸)/sinÂ Î¸ cos Î¸]

= [(sin Î¸-cos Î¸)(sin^{2}Î¸+cos^{2}Î¸+sinÂ Î¸ cos Î¸)]/[(sin Î¸-cos Î¸)sinÂ Î¸ cos Î¸]

= (1Â + sinÂ Î¸ cos Î¸)/sinÂ Î¸ cos Î¸

= 1/sinÂ Î¸ cos Î¸Â + 1

= 1Â + sec Î¸ cosec Î¸ = R.H.S.

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

Hence proved

(iv) Â (1Â + sec A)/sec A = sin^{2}A/(1-cos A)

First find the simplified form of L.H.S

L.H.S. = (1Â + sec A)/sec A

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

= (1Â + 1/cos A)/1/cos A

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

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

R.H.S. = sin^{2}A/(1-cos A)

We know that sin^{2}A = (1 â€“ cos^{2}A), we get

= (1 â€“ cos^{2}A)/(1-cos A)

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

Therefore,Â sin^{2}A/(1-cos A)= cos AÂ + 1

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

Hence proved

(v) (cos Aâ€“sin A+1)/(cos A+sin Aâ€“1) =Â cosec A + cot A, using the identity cosec^{2}A = 1+cot^{2}A.

With the help of identity function, cosec^{2}A = 1+cot^{2}A, let us prove the above equation.

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

Divide the numerator and denominator by sin A, we get

= (cos Aâ€“sin A+1)/sin A/(cos A+sin Aâ€“1)/sin A

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

= (cot A â€“ 1 +Â cosec A)/(cot A+ 1 â€“ cosec A)

= (cot A â€“ cosec^{2}AÂ + cot^{2}A +Â cosec A)/(cot A+ 1 â€“ cosec A) (using cosec^{2}A â€“ cot^{2}A = 1

= [(cot A +Â cosec A) â€“ (cosec^{2}AÂ â€“ cot^{2}A)]/(cot A+ 1 â€“ cosec A)

= [(cot A +Â cosec A) â€“Â (cosec AÂ + cot A)(cosec AÂ â€“ cot A)]/(1 â€“ cosec A +Â cot A)

= Â (cot AÂ +Â cosec A)(1 â€“ cosec AÂ +Â cot A)/(1 â€“ cosec AÂ +Â cot A)

= Â cot AÂ +Â cosec A = R.H.S.

Therefore, (cos Aâ€“sin A+1)/(cos A+sin Aâ€“1) =Â 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,

= âˆš(sec A+ tan A)/(sec A-tan A)

Now using rationalization, we get

= (sec A + tan A)/1

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

Hence proved

(vii) (sin Î¸ â€“ 2sin^{3}Î¸)/(2cos^{3}Î¸-cos Î¸) = tan Î¸

L.H.S. = (sin Î¸ â€“ 2sin^{3}Î¸)/(2cos^{3}Î¸ â€“ cos Î¸)

Take sin Î¸ as in numerator and cos Î¸ in denominator as outside, it becomes

= [sin Î¸(1 â€“ 2sin^{2}Î¸)]/[cos Î¸(2cos^{2}Î¸- 1)]

We know that sin^{2}Î¸ = 1-cos^{2}Î¸

= sin Î¸[1 â€“ 2(1-cos^{2}Î¸)]/[cos Î¸(2cos^{2}Î¸ -1)]

=Â [sin Î¸(2cos^{2}Î¸ -1)]/[cos Î¸(2cos^{2}Î¸ -1)]

= tan Î¸ = R.H.S.

Hence proved

(viii) (sin AÂ + cosec A)^{2Â }+ (cos A + sec A)^{2}Â =Â 7+tan^{2}A+cot^{2}A

L.H.S. = (sin AÂ + cosec A)^{2Â }+ (cos A + sec A)^{2}

It is of the form (a+b)^{2}, expand it

(a+b)^{2} =a^{2} + b^{2} +2ab

^{Â Â Â Â Â Â Â Â }= (sin^{2}AÂ + cosec^{2}AÂ + 2 sin A cosec A)Â + (cos^{2}AÂ +Â sec^{2}AÂ + 2 cos A sec A)

= (sin^{2}A + cos^{2}A) + 2 sin A(1/sin A)Â + 2 cos A(1/cos A)Â + 1 + tan^{2}A + 1Â + cot^{2}A

= 1Â + 2Â + 2Â + 2 + tan^{2}AÂ +Â cot^{2}A

= 7+tan^{2}A+cot^{2}A = R.H.S.

Therefore, (sin AÂ + cosec A)^{2Â }+ (cos A + sec A)^{2}Â =Â 7+tan^{2}A+cot^{2}A

Hence proved.

(ix) (cosec A â€“ sin A)(sec A â€“ cos A) = 1/(tan A+cotA)

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

L.H.S. = (cosec A â€“ sin A)(sec A â€“ cos A)

Now, substitute the inverse and equivalent trigonometric ratio forms

= (1/sin A â€“ sin A)(1/cos A â€“ cos A)

= [(1-sin^{2}A)/sin A][(1-cos^{2}A)/cos A]

= (cos^{2}A/sin A)Ã—(sin^{2}A/cos A)

= cos A sin A

Now, simplify the R.H.S

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

= 1/(sin A/cos AÂ +cos A/sin A)

= 1/[(sin^{2}A+cos^{2}A)/sin A cos A]

= cos A sin A

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

(cosec A â€“ sin A)(sec A â€“ cos A) = 1/(tan A+cotA)

Hence proved

(x) Â (1+tan^{2}A/1+cot^{2}A) = (1-tan A/1-cot A)^{2}Â =^{Â }tan^{2}A

L.H.S. = (1+tan^{2}A/1+cot^{2}A)

Since cot function is the inverse of tan function,

= (1+tan^{2}A/1+1/tan^{2}A)

= 1+tan^{2}A/[(1+tan^{2}A)/tan^{2}A]

Now cancel the 1+tan^{2}A terms, we get

= tan^{2}A

(1+tan^{2}A/1+cot^{2}A) = tan^{2}A

Similarly,

(1-tan A/1-cot A)^{2}Â =^{Â }tan^{2}A

Hence proved

Exercise 8.4 of 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 cos and sin function, the trigonometric identities of other functions are easily found since the functions are inter-related 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 NCERT solutions for class 10 maths chapter 8 to solve more problems to score high in the examinations.