Home | | Maths 10th Std | Trigonometric identities

Chapter: 10th Mathematics : Trigonometry

Trigonometric identities

For all real values of θ , we have the following three identities.

Trigonometric identities

For all real values of θ , we have the following three identities.

(i) sin 2 θ + cos2 θ=1

(ii) 1 + tan2 θ = sec2 θ

(iii) 1 + cot2 θ = cosec2 θ

These identities are termed as three fundamental identities of trigonometry. We will now prove them as follows.



These identities can also be rewritten as follows.


Note

Though the above identities are true for any angle θ, we will consider the six trigonometric ratios only for 0° < θ < 90°

 

Example 6.1

Prove that tan 2 θ − sin2 θ = tan 2 θ sin2 θ

Solution

tan 2 θ - sin2 θ = tan2 θ − . cos2 θ

 = tan 2 θ(1 − cos 2 θ) = tan 2 θ sin2 θ

 

Example 6.2

Prove that 

Solution


 

Example 6.3

Prove that 1 +  = cosec θ

Solution


 

Example 6.4

Prove that sec θ − cos θ = tan θ sin θ

Solution


 

Example 6.5 Prove that  = cosec θ + cot θ

Solution


 

Example 6.6

Prove that  = cot θ

Solution


 

Example 6.7

Prove that sin 2 A cos2 B + cos 2 A sin2 B + cos 2 A cos2 B + sin 2 A sin2 B = 1

Solution

sin 2 A cos2 B + cos 2 A sin2 B + cos 2 A cos2 B + sin 2 A sin2 B

 = sin 2 A cos2 B + sin 2 A sin2 B + cos 2 A sin2 B + cos 2 A cos2

 = sin2 A(cos2 B + sin2 B ) + cos2 A(sin2 B + cos 2 B)

 = sin 2 A(1) + cos2 A(1)              (since  sin2 B + cos 2 B = 1)

 = sin 2 A + cos2 A = 1

 

Example 6.8

If cos θ + sin θ = √2 cos θ, then prove that cos θ − sin θ =  √2sin θ

Solution

Now, cos θ + sin θ = √2 cos θ

Squaring both sides,

(cos θ + sin θ)2 = (√2 cos θ)2

cos 2 θ + sin2 θ + 2 sin θ cos θ = 2 cos2 θ

2 cos2 θ - cos2 θ - sin2 θ = 2 sin θ cos θ

cos 2 θ - sin2 θ = 2 sin θ cos θ

(cos θ + sin θ)(cos θ − sin θ) =2 sin θ cos θ


Therefore cos θ − sin θ = √2sin θ

 

Example 6.9 

Prove that (cosec θ − sin θ)(sec θ − cos θ)(tan θ + cot θ) =1

Solution

(cosec θ sin θ)(sec θ cos θ)(tan θ + cot θ)


 

Example 6.10

Prove that 

Solution 


 

Example 6.11 

If cosec θ +cot θ = P , then prove that cos θ = 

Solution 

Given cosec θ +cot θ = P   ...(1)

cosec2 θ - cot2 θ =1 (identity)

cosec θ - cot θ = 1 / (cosec θ + cot θ)


 

Example 6.12

Prove that tan2A tan2 B =  

Solution


 

Example 6.13


Solution


 

Example 6.14 Prove that 

Solution


 

Example 6.15


Solution


 

Example 6.16

Prove that  = sin 2 A cos2 A

Solution


 

Example 6.17


Solution

 


Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
10th Mathematics : Trigonometry : Trigonometric identities |


Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.