MQ62 --- Trigonometry

csc 2x
= 1 / sin2x
= 1 / (2sinx cosx)
= [2cos²x - (2cos²x - 1)] / (2 sin x cos x)
= cos x / sin x - (2cos²x - 1) / (2 sin x cos x)
= cos x / sin x - cos 2x / sin 2x
= cot x - cot 2x Similarly ,
csc 4x = cot 2x - cot 4x
csc 8x = cot 4x - cot 8x
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csc 2ⁿx = cot 2ⁿ⁻¹x - cot 2ⁿx
∴ Σ(k = 1 to n) csc 2ᵏx
= csc 2x + csc 4x + csc 8x + ... + csc 2ⁿx
= (cot x - cot 2x) + (cot 2x - cot 4x) + (cot 4x - 8x) + ... + (cot 2ⁿ⁻¹x - cot 2ⁿx)
= cot x - cot 2ⁿx