Calculate the value tan 9° - tan 27°- tan 63° + tan 81°

Given

tan 9° – tan 27°- tan 63° + tan 81°

Find out

We need to evaluate the given trigonometric function

Solution

tan 9° – tan 27°- tan 63° + tan 81°

Combing all like terms together

= tan 9° + tan 81° -tan 27°- tan 63°

= tan 9° + tan (90°-9°) – tan 27° – tan (90°- 27°)

= tan 9° + cot 9° -(tan 27°+ cot 27°) ……(1)

We can express it as

tan 9°+ cot 9° = 1/ sin 9°cos 9°

= 2/ sin 18° ……………(2)

tan 27° + cot 27° = 1/ sin 27° cos 27°

= 2/ sin 54°

= 2/ cos 36° ………..(3)

Substitute (2) and (3) in (1), we get

tan 9° + cot 9° -(tan 27°+ cot 27°) = (2/ sin 18° )- (2/ cos 36°)

= 4

Answer

Therefore, the value of tan 9° – tan 27°- tan 63° + tan 81° = 4

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