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Question
Find the value of tan 9° – tan 27° – tan 63° + tan 81°
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Solution
We have tan 9° – tan 27° – tan 63° + tan 81°
= 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)
Also tan 9° + cot 9° = `1/(sin 9^circ cos 9^circ)`
= `2/(sin 18^circ)` .....(2)
Similarly, tan 27° + cot 27° = `1/(sin 27^circ cos 27^circ)`
= `2/sin54^circ`
= `2/cos36^circ` .....(3)
Using (2) and (3) in (1), we get
tan 9° – tan 27° – tan 63° + tan 81° = `2/(sin18^circ) - 2/(cos36^circ)`
= `(2 xx 4)/(sqrt(5) - 1)`
= `(2 xx 4)/(sqrt(5) + 1)`
= 4
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