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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 |

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