Q.181. Trigonometry

Question 181.
Express (1-cot2010°)(1-cot1995°) as a single real number.

Answer 181.
2010° = [(11) * (180) + 30]° and
1995° = [(11) * (180) + 15]°
=> (1 - cot2010°) (1 - cot1995°)
= (1 - cot30°) (1 - cot15°)
= (1 - √3) (1 - cot15°)
 cot15°
= √[(1 + cos30°) / (1 - cos30°)
= √[(2 + √3)/(2 - √3)]
= 2 + √3 ... (by rationalising)

=> (1 - cot2010°) (1 - cot1995°)
= (1 - √3) [1 - (2 + √3)]
= (√3 - 1) (√3 + 1)
= 2.

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