Question
If tan 15° = 2 –√3, what is the value of tan 15º
cot 75°+ tan 75º cot 15°.Solution
tan 15° × cot 75° + tan 75° × cot 15° = tan 15° × tan (90° - 15°) + cot (90° -75°) × cot 15° = tan 15° × tan 15° + cot 15° × cot 15° = (tan15°)2 + (cot 15°)2  Now, given tan 15° = 2 - √3 And cot 15° = 1/tan15° = 1/ (2 - √3) = 2 + √3        [on rationalizing]  Thus, tan 15° × cot 75° + tan 75° × cot 15°  = (tan15°)2 + (cot 15°)2 = (2 - √3)2 + (2 + √3)2 = 7 – 4√3 + 7 + 4√3 = 7 + 7 = 14
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