# If cosec a +cot a =q

1## Answers

Given : cosec a+cot a = q ---------> 1

We know that cosec²Ф-cot²Ф = 1

(cosec a+cot a)+(cosec a-cot a) = 1

cosec a-cot a = 1/(cosec a+cot a)

cosec a-cot a = 1/q -------> 2

∴ Hence proved.

Equation 1 & Equation 2

cosec a+cot a = q

cosec a-cot a = 1/q

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2cosec a +0 = q+1/q

2cosec a = (q²+1)/q

cosec a = (q²+1)/2q

(1/sin a) = (q²+1)/2q

sin a = 2q/(q²+1)

∴sin a = 2q/(q²+1)

sec a = ?

We know that sin²Ф+cos²Ф = 1

[2q/(q²+1)]²+cos²Ф =1

cos² a = 1-[2q/(q²+1)]²

cos² a = 1-4q²/(q²+1)²

cos² a = [(q²)²+1²+2.q².1-4q²]/(q²+1)²

cos² a = [(q²)²+1²-2.q².1]/(q²+1)²

cos² a = (q²-1)/(q²+1)

∴cos² a = (q²-1)/(q²+1)