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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

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10th Mathematics : UNIT 6 : Trigonometry : Trigonometric identities |