The branch of mathematics which deals with the measurement of the sides and the angles of a triangle is trigonometry. We know that by now this topic would already seem difficult and complicated as its completely new to you. So, in order to make your learning process smooth and hassle-free the RD Sharma Solutions prepared by our expert team at BYJU’S will help students get the correct understanding of various chapters in the book.

Trigonometric Identities is the 6th chapter of RD Sharma Class 10 which has two exercises and its solved answers with detailed explanations are given here RD Sharma Solutions for Class 10. The previous chapter was about trigonometric ratios and relations between them. But this chapter will be about proving some trigonometric identities and use them to prove other useful trigonometric identities.

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### Access the RD Sharma Solutions For Class 10 Chapter 6 – Trigonometric Identities

Exercise 6.1 Page No: 6.43

**Prove the following trigonometric identities: **

**1. (1 â€“ cos ^{2} A) cosec^{2} A = 1 **

**Solution: **

Taking the L.H.S,

(1 â€“ cos^{2} A) cosec^{2} A

= (sin^{2} A) cosec^{2} A [âˆµ sin^{2} A + cos^{2 }A = 1 â‡’1 â€“ sin^{2} A = cos^{2} A]

= 1^{2}

^{ }= 1 = R.H.S

– Hence Proved

**2. (1 + cot ^{2}Â A) sin^{2}Â A = 1Â **

**Solution:Â **

By using the identity,

cosec^{2 }A – cot^{2}Â A = 1 â‡’ cosec^{2 }A = cot^{2}Â A + 1

Taking,

L.H.S = (1 + cot^{2}Â A) sin^{2}Â A

= cosec^{2}Â A sin^{2}Â A

= (cosec A sin A)^{2}Â

= ((1/sin A) Ã— sin A)^{2}Â

= (1)^{2}Â

= 1Â

= R.H.S

– Hence Proved

**3. tan ^{2 }Î¸ cos^{2 }Î¸ =Â 1 âˆ’ cos^{2 }Î¸Â **

**Solution:Â **

We know that,

Â sin^{2Â }Î¸ + cos^{2Â }Î¸ = 1Â

Taking,

L.H.S =Â tan^{2Â }Î¸ cos^{2Â }Î¸

= (tan Î¸ Ã— cos Î¸)^{2}

= (sin Î¸)^{2}Â

= sin^{2 }Î¸Â

= 1 – cos^{2 }Î¸Â

= R.H.S

– Hence Proved

**4. cosec Î¸ âˆš(1 â€“ cos ^{2} Î¸) = 1**

**Solution: **

Using identity,

sin^{2Â }Î¸ + cos^{2Â }Î¸ = 1Â â‡’ sin^{2Â }Î¸ = 1 – cos^{2Â }Î¸

Taking L.H.S,

L.H.S = cosec Î¸ âˆš(1 â€“ cos^{2} Î¸)

= cosec Î¸ âˆš( sin^{2Â }Î¸)

= cosec Î¸ x sin^{Â }Î¸

= 1

= R.H.S

– Hence Proved

**5. (sec ^{2 }Î¸ âˆ’ 1)(cosec^{2 }Î¸ âˆ’ 1) = 1Â **

**Solution:**

Using identities,Â

(sec^{2 }Î¸ âˆ’ tan^{2 }Î¸) = 1Â and (cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸) = 1

We have,

L.H.S =Â (sec^{2 }Î¸ – 1)(cosec^{2}Î¸ – 1)

= tan^{2}Î¸ Ã— cot^{2}Î¸

= (tan Î¸ Ã— cot Î¸)^{2}

= (tan Î¸ Ã— 1/tan Î¸)^{2}

= 1^{2}

= 1

= R.H.S

– Hence Proved

**6. tan Î¸ + 1/ tan Î¸ = sec Î¸ cosec Î¸**

**Solution: **

We have,

L.H.S = tan Î¸ + 1/ tan Î¸

= (tan^{2} Î¸ + 1)/ tan Î¸

= sec^{2} Î¸ / tan Î¸ [âˆµ sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1]

= (1/cos^{2} Î¸) x 1/ (sin Î¸/cos Î¸) [âˆµ tan^{ }Î¸ = sin Î¸ / cos Î¸]

= cos Î¸/ (sin Î¸ x cos^{2} Î¸)

= 1/ cos Î¸ x 1/ sin Î¸

= sec Î¸ x cosec Î¸

= sec Î¸ cosec Î¸

= R.H.S

– Hence Proved

**7. cos Î¸/ (1 – sin Î¸) = (1 + sin Î¸)/ cos Î¸ **

**Solution: **

We know that,Â

sin^{2 }Î¸ + cos^{2 }Î¸ = 1

So, by multiplying both the numerator and the denominator by (1+ sin Î¸), we get

L.H.S =

= R.H.S

– Hence Proved

8. **cos Î¸/ (1 + sin Î¸) = (1 – sin Î¸)/ cos Î¸ **

**Solution: **

We know that,Â

sin^{2 }Î¸ + cos^{2 }Î¸ = 1

So, by multiplying both the numerator and the denominator by (1- sin Î¸), we get

L.H.S =

= R.H.S

– Hence Proved

**9. cos ^{2 }Î¸ + 1/(1 + cot^{2 }Î¸) = 1**

**Solution: **

** **We already know that,

cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸ = 1 and sin^{2 }Î¸ + cos^{2 }Î¸ = 1

Taking L.H.S,

** **

= cos^{2} A + sin^{2} A

= 1

= R.H.S

– Hence Proved

**10. sin ^{2 }A + 1/(1 + tan ^{2 }A) = 1**

**Solution: **

** **We already know that,

sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1Â and sin^{2 }Î¸ + cos^{2 }Î¸ = 1

Taking L.H.S,

= sin^{2} A + cos^{2} A

= 1

= R.H.S

– Hence Proved

**11.**

**Solution: **

** **We know that, sin^{2 }Î¸ + cos^{2 }Î¸ = 1

Taking the L.H.S,

= R.H.S

– Hence Proved

**12. 1 â€“ cos Î¸/ sin Î¸ = sin Î¸/ 1 + cos Î¸ **

**Solution: **

We know that,Â

sin^{2 }Î¸ + cos^{2 }Î¸ = 1

So, by multiplying both the numerator and the denominator by (1+ cos Î¸), we get

= R.H.S

– Hence Proved

**13.** **sin Î¸/ (1 â€“ cos Î¸) = cosec Î¸ + cot Î¸ **

**Solution: **

** **

Taking L.H.S,

= cosec Î¸ + cot Î¸

= R.H.S

– Hence Proved

**14. (1 â€“ sin Î¸) / (1 + sin Î¸) = (sec Î¸ â€“ tan Î¸) ^{2}**

**Solution: **

Taking the L.H.S,

= (sec Î¸ – tan Î¸)^{2}

_{ } = R.H.S

– Hence Proved

**15. **

**Solution: **

Taking L.H.S,

= cot Î¸

= R.H.S

– Hence Proved

**16. tan ^{2 }Î¸ âˆ’ sin^{2 }Î¸Â =Â tan^{2 }Î¸ sin^{2 }Î¸Â **

**Solution: **

Taking L.H.S,

L.H.S = tan^{2 }Î¸ âˆ’ sin^{2 }Î¸**Â **

= tan^{2 }Î¸ sin^{2 }Î¸Â

= R.H.S

– Hence Proved

**17. (cosecÂ Î¸Â + sinÂ Î¸)(cosecÂ Î¸Â – sinÂ Î¸) =Â cot ^{2}Î¸ + cos^{2}Î¸Â **

**Solution: **

Taking L.H.S = (cosecÂ Î¸Â + sinÂ Î¸)(cosecÂ Î¸Â – sinÂ Î¸)

On multiplying we get,

= cosec^{2}Â Î¸Â â€“ sin^{2}Â Î¸

=Â (1 + cot^{2}Â Î¸)Â – (1 – cos^{2}Â Î¸) [Using cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸ = 1Â and sin^{2 }Î¸ + cos^{2 }Î¸ = 1]

=Â 1 + cot^{2}Â Î¸Â – 1 + cos^{2}Â Î¸Â

= cot^{2}Â Î¸Â + cos^{2}Â Î¸Â

= R.H.SÂ Â Â

– Hence Proved

**18. (sec Î¸Â + cosÂ Î¸) (secÂ Î¸Â – cosÂ Î¸) =Â tan ^{2 }Î¸ + sin^{2 }Î¸Â **

**Solution: **

Taking L.H.S = (secÂ Î¸Â + cosÂ Î¸)(secÂ Î¸Â – cosÂ Î¸)

On multiplying we get,

= sec^{2}Â Î¸Â â€“ sin^{2}Â Î¸

= (1 + tan^{2}Â Î¸)Â – (1 – sin^{2}Â Î¸) [Using sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1Â and sin^{2 }Î¸ + cos^{2 }Î¸ = 1]

= 1 + tan^{2}Â Î¸Â – 1 + sin^{2}Â Î¸Â

=Â tan ^{2 }Î¸Â + sin ^{2}Â Î¸Â

= R.H.S

– Hence Proved

**19. sec A(1- sin A) (sec A + tan A) = 1**

**Solution: **

Taking L.H.S = sec A(1 â€“ sin A)(sec A + tan A)

Substituting sec A = 1/cos A and tan A =sin A/cos A in the above we have,

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

= 1 â€“ sin^{2} A / cos^{2} A [After taking L.C.M]

= cos^{2} A / cos^{2} A [âˆµ 1 â€“ sin^{2} A = cos^{2} A]

= 1

= R.H.S

– Hence Proved

**20. (cosec A â€“ sin A)(sec A â€“ cos A)(tan A + cot A) = 1Â **

**Solution: **

Taking L.H.S = (cosec A â€“ sin A)(sec A â€“ cos A)(tan A + cot A)

** **

** ** = (cos^{2} A/ sin A) (sin^{2} A/ cos A) (1/ sin A cos A) [âˆµ sin^{2 }Î¸ + cos^{2 }Î¸ = 1]

= sin A x cos A x (1/ cos A sin A)

= R.H.S

– Hence Proved

**21. (1 +Â tan ^{2 }Î¸)(1 – sinÂ Î¸)(1 + sinÂ Î¸) = 1**

**Solution: **

Taking L.H.S = (1 +Â tan^{2}Î¸)(1 – sin Î¸)(1 + sin Î¸)

And, we know sin^{2}Â Î¸Â + cos^{2}Â Î¸Â = 1 and sec^{2}Â Î¸Â – tan^{2}Â Î¸Â = 1

So,

L.H.S = (1 +Â tan^{2} Î¸)(1 – sinÂ Î¸)(1 + sinÂ Î¸)

= (1 +Â tan^{2 }Î¸){(1 – sinÂ Î¸)(1 + sinÂ Î¸)}

= (1 +Â tan^{2 }Î¸)(1 –Â sin^{2 }Î¸)

= sec^{2 }Î¸ (cos^{2} Î¸)

= (1/ cos^{2} Î¸) x cos^{2} Î¸

= 1

= R.H.S

– Hence Proved

**22. sin ^{2 }A cot^{2 }A + cos^{2 }A tan^{2 }AÂ = 1**

**Solution: **

We know that,

cot^{2 }A = cos^{2 }A/ sin^{2} A and tan^{2} A = sin^{2} A/cos^{2} A

Substituting the above in L.H.S, we get

L.H.S = sin^{2 }A cot^{2 }A + cos^{2 }A tan^{2 }AÂ

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

= cos^{2 }A + sin^{2} A

= 1 [âˆµ sin^{2 }Î¸ + cos^{2 }Î¸ = 1]

= R.H.S

– Hence Proved

**23. **

**Solution: **

(i) Taking the L.H.S and using sin^{2 }Î¸ + cos^{2 }Î¸ = 1, we have

L.H.S = cot Î¸ â€“ tan Î¸

= R.H.S

– Hence Proved

(ii) Taking the L.H.S and using sin^{2 }Î¸ + cos^{2 }Î¸ = 1, we have

L.H.S = tan Î¸ â€“ cot Î¸

= R.H.S

– Hence Proved

**24. (cos ^{2} Î¸/ sin Î¸) â€“ cosec Î¸ + sin Î¸ = 0**

**Solution: **

Taking L.H.S and using sin^{2 }Î¸ + cos^{2 }Î¸ = 1, we have

= â€“ sin Î¸ + sin Î¸

= 0

= R.H.S

- Hence proved

**25. **

**Solution: **

** **Taking L.H.S,

** **

** ** = 2 sec^{2} A

= R.H.S

- Hence proved

**26. **

**Solution:**

Taking the LHS and using sin^{2 }Î¸ + cos^{2 }Î¸ = 1, we have

= 2/ cos Î¸

= 2 sec Î¸

= R.H.S

- Hence proved

**27. **

**Solution: **

** **

Taking the LHS and using sin^{2 }Î¸ + cos^{2 }Î¸ = 1, we have

= R.H.S

**28. **

**Solution: **

Taking L.H.S,

Using sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1Â and cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸ = 1

= R.H.S

And, taking

=

[Using sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1Â and cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸ = 1]

=

=

= tan^{2 }Î¸ = R.H.S

**29. **

**Solution: **

Taking L.H.S and using sin^{2 }Î¸ + cos^{2 }Î¸ = 1, we have

= R.H.S

**30. **

**Solution: **

Taking LHS, we have

= 1 + tan Î¸ + cot Î¸

= R.H.S

**31. sec ^{6 }Î¸ = tan^{6 }Î¸ + 3 tan^{2Â }Î¸ sec^{2 }Î¸ + 1**

**Solution:Â **

From trig. Identities we have,

sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1Â

On cubingÂ both sides,

(sec^{2}Î¸ âˆ’ tan^{2}Î¸)^{3 }= 1

sec^{6 }Î¸ âˆ’ tan^{6 }Î¸ âˆ’ 3sec^{2 }Î¸ tan^{2 }Î¸(sec^{2 }Î¸ âˆ’ tan^{2 }Î¸) = 1Â

^{3}Â = a

^{3}Â – b

^{3}â€“ 3ab(a – b)]Â

sec^{6 }Î¸ âˆ’ tan^{6 }Î¸ âˆ’ 3sec^{2 }Î¸ tan^{2 }Î¸ = 1

â‡’ sec^{6 }Î¸ = tan^{6 }Î¸ + 3sec^{2 }Î¸ tan^{2 }Î¸ + 1

Hence, L.H.S = R.H.S

**32. cosec ^{6 }Î¸ = cot^{6 }Î¸ + 3cot^{2 }Î¸ cosec^{2 }Î¸ + 1**

**Solution:**

From trig. Identities we have,

cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸ = 1Â

On cubingÂ both sides,

(cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸)^{3}Â = 1

cosec^{6 }Î¸ âˆ’ cot^{6 }Î¸ âˆ’ 3cosec^{2 }Î¸ cot^{2 }Î¸ (cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸) = 1

^{3}Â = a

^{3}Â – b

^{3}â€“ 3ab(a – b)]

cosec^{6 }Î¸ âˆ’ cot^{6 }Î¸ âˆ’ 3cosec^{2 }Î¸ cot^{2 }Î¸ = 1

â‡’ cosec^{6 }Î¸ = cot^{6 }Î¸ + 3 cosec^{2 }Î¸ cot^{2 }Î¸ + 1Â

Hence, L.H.S = R.H.S

- Hence proved

**33. **

**Solution: **

Taking L.H.S and using sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1Â â‡’ 1 + tan^{2 }Î¸ = sec^{2 }Î¸

= R.H.S

- Hence proved

**34. **

**Solution: **

Taking L.H.S and using the identity sin^{2}A + cos^{2}AÂ = 1, we get

sin^{2}A = 1 âˆ’ cos^{2}AÂ

â‡’ sin^{2}A = (1 â€“ cos A)(1 + cos A)

- Hence proved

**35. **

**Solution: **

We have,

** **

Rationalizing the denominator and numerator with (sec A + tan A) and using sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1Â we get,

= R.H.S

- Hence proved

**36. **

**Solution: **

We have,

On multiplying numerator and denominator by (1 â€“ cos A), we get

** **

** ** = R.H.S

- Hence proved

**37. (i) **

**Solution: **

Taking L.H.S and rationalizing the numerator and denominator with âˆš(1 + sin A), we get

** **

** ** = R.H.S

- Hence proved

** (ii) **

**Solution: **

Taking L.H.S and rationalizing the numerator and denominator with its respective conjugates, we get

= 2 cosec A

= R.H.S

- Hence proved

**38. Prove that: **

**(i) **

**Solution: **

Taking L.H.S and rationalizing the numerator and denominator with its respective conjugates, we get

= R.H.S

- Hence proved

**(ii) **

**Solution:**

Taking L.H.S and rationalizing the numerator and denominator with its respective conjugates, we get

** **= R.H.S

- Hence proved

**(iii) **

**Solution: **

Taking L.H.S and rationalizing the numerator and denominator with its respective conjugates, we get

** **

** **

** **= 2 cosec Î¸

= R.H.S

- Hence proved

**(iv) **

**Solution: **

Taking L.H.S, we have

** **

= R.H.S

- Hence proved

**39. **

**Solution: **

** **

** **Taking LHS = (sec A â€“ tan A)^{2} , we have

= R.H.S

- Hence proved

**40. **

**Solution: **

Taking L.H.S and rationalizing the numerator and denominator with (1 â€“ cos A), we get

** **

** ** = (cosec A – cot A)^{2}Â

= (cot A – cosec)^{2}Â

= R.H.S

- Hence proved

**41. **

**Solution: **

Considering L.H.S and taking L.C.M and on simplifying we have,

= 2 cosec A cot A = RHS

- Hence proved

**42. **

**Solution: **

Taking LHS, we have

= cos A + sin A

= RHS

- Hence proved

**43. **

**Solution: **

Considering L.H.S and taking L.C.M and on simplifying we have,

= 2 sec^{2 }A

= RHS

- Hence proved

Exercise 6.2 Page No: 6.54

**1. IfÂ cos Î¸ = 4/5, find all other trigonometric ratios of angleÂ Î¸.Â **

**Solution: **

** **

** **We have,

cos Î¸ = 4/5

And we know that,

sin Î¸ = âˆš(1 – cos^{2 }Î¸)

â‡’ sin Î¸ = âˆš(1 â€“ (4/5)^{2})

= âˆš(1 â€“ (16/25))

= âˆš[(25 â€“ 16)/25]

= âˆš(9/25)

= 3/5

âˆ´ sin Î¸ = 3/5

Since, cosec Î¸ = 1/ sin Î¸

= 1/ (3/5)

â‡’ cosec Î¸ = 5/3

And, sec Î¸ = 1/ cos Î¸

= 1/ (4/5)

â‡’ cosec Î¸ = 5/4

Now,

tan Î¸ = sin Î¸/ cos Î¸

= (3/5)/ (4/5)

â‡’ tan Î¸ = 3/4

And, cot Î¸ = 1/ tan Î¸

= 1/ (3/4)

â‡’ cot Î¸ = 4/3

**2. IfÂ sin Î¸ = 1/âˆš2, find all other trigonometric ratios of angleÂ Î¸.**

**Solution: **

** **

We have,

sin Î¸ = 1/âˆš2

And we know that,

cos Î¸ = âˆš(1 – sin^{2 }Î¸)

â‡’ cos Î¸ = âˆš(1 â€“ (1/âˆš2)^{2})

= âˆš(1 â€“ (1/2))

= âˆš[(2 â€“ 1)/2]

= âˆš(1/2)

= 1/âˆš2

âˆ´ cos Î¸ = 1/âˆš2

Since, cosec Î¸ = 1/ sin Î¸

= 1/ (1/âˆš2)

â‡’ cosec Î¸ = âˆš2

And, sec Î¸ = 1/ cos Î¸

= 1/ (1/âˆš2)

â‡’ cosec Î¸ = âˆš2

Now,

tan Î¸ = sin Î¸/ cos Î¸

= (1/âˆš2)/ (1/âˆš2)

â‡’ tan Î¸ = 1

And, cot Î¸ = 1/ tan Î¸

= 1/ (1)

â‡’ cot Î¸ = 1

**3. **

**Solution: **

Given,

tan Î¸ = 1/âˆš2

By using sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1,

**4. **

**Solution: **

Given,

tan Î¸ = 3/4

By using sec^{2 }Î¸ âˆ’ tan^{2 }Î¸ = 1,

** **

** sec **Î¸ = 5/4

Since, sec Î¸ = 1/ cos Î¸

â‡’ cos Î¸ = 1/ sec Î¸

= 1/ (5/4)

= 4/5** **

** **

So,

** **

**5. **

**Solution: **

Given, tan Î¸ = 12/5

Since, cot Î¸ = 1/ tan Î¸ = 1/ (12/5) = 5/12

Now, by using cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸ = 1

cosec Î¸ = âˆš(1 + cot^{2 }Î¸)

= âˆš(1 + (5/12)^{2 })

= âˆš(1 + 25/144)

= âˆš(169/ 25)

â‡’ cosec Î¸ = 13/5

Now, we know that

sin Î¸ = 1/ cosec Î¸

= 1/ (13/5)

â‡’ sin Î¸ = 5/13

Putting value of sin Î¸ in the expression we have,

= 25/ 1

= 25

**6. **

**Solution: **

**Given, **

cot Î¸ = 1/âˆš3

Using cosec^{2 }Î¸ âˆ’ cot^{2 }Î¸ = 1, we can find cosec Î¸

cosec Î¸ = âˆš(1 + cot^{2} Î¸)

= âˆš(1 + (1/âˆš3)^{2})

= âˆš(1 + (1/3)) = âˆš((3 + 1)/3)

= âˆš(4/3)

â‡’ cosec Î¸ = 2/âˆš3

So, sin Î¸ = 1/ cosec Î¸ = 1/ (2/âˆš3)

â‡’ sin Î¸ = âˆš3/2

And, we know that

cos Î¸ = âˆš(1 – sin^{2} Î¸)

= âˆš(1 â€“ (âˆš3/2)^{2})

= âˆš(1 â€“ (3/4))

** = **âˆš((4 â€“ 3)/4)

= âˆš(1/4)

â‡’ cos Î¸ = 1/2

Now, using cos Î¸ and sin Î¸ in the expression, we have

= 3/5

**7. **

**Solution: **

** **Given,

cosec A = âˆš2

Using cosec^{2 }A âˆ’ cot^{2 }A = 1, we find cot A