tan(9 theta)-tan(27 theta)-tan(63 theta)+ tan(81 theta)=4

Question

We need to prove the above.

Answer 1

 

tan 9 – tan 27 – tan 63 + tan 81

= tan 9 + tan 81 – (tan 27 + tan 63)

= tan (9 + 81) (1 + tan 9 × tan 81) – tan (27 + 63) (1 + tan 27 × tan 63)

= tan 90 (1 + tan 9 × cot (90 – 81)) – tan 90 (1 + tan 27 × cot (90 – 63))

= tan 90 [1 + tan 9 × cot 9 – (1 + tan 27 cot 27)]

= tan 90 [1 + 1 – (1 + 1)] = tan 90 [2 – 2] = tan 90 × 0 = 0

Answer 2

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