JEEOlympiad
Compute the exact value of tan 9° − tan 27° − tan 63° + tan 81°. Give…
Question
Compute the exact value of tan 9° − tan 27° − tan 63° + tan 81°. Give a single integer.
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Step-by-step solution
Group as (tan 9° + tan 81°) − (tan 27° + tan 63°). Note 81° = 90° − 9°, so tan 81° = cot 9°; and 63° = 90° − 27°, so tan 63° = cot 27°.
tan θ + cot θ = sin θ/cos θ + cos θ/sin θ = 1/(sin θ cos θ) = 2/sin 2θ.
So tan 9° + tan 81° = tan 9° + cot 9° = 2/sin 18°, and tan 27° + tan 63° = tan 27° + cot 27° = 2/sin 54°.
The expression = 2/sin 18° − 2/sin 54°. Using sin 18° = (√5 − 1)/4 and sin 54° = (√5 + 1)/4:
2/sin 18° = 8/(√5 − 1) = 2(√5 + 1); 2/sin 54° = 8/(√5 + 1) = 2(√5 − 1).
Difference = 2(√5 + 1) − 2(√5 − 1) = 4.
Final answer4
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