Question

if xcos theta=y prove that cosec theta +cot theta =√x+y÷x-y

Answer 1

xcosθ=y
or, cosθ=y/x
∵, sin²θ+cos²θ=1
or, sin²θ=1-cos²θ
or, sin²θ=1-y²/x²
or, sin²θ=(x²-y²)/x²
∴, sinθ=√(x²-y²)/x
∴, cosecθ=1/sinθ=x/√(x²-y²) and
cotθ=cosθ/sinθ=(y/x)/{√(x²-y²)/x}=y/√(x²-y²)
∴, cosecθ+cotθ
=x/√(x²-y²)+y/√(x²-y²)
=(x+y)/√(x²-y²)
=(x+y)/√(x+y)(x-y)
=√(x+y)/(x-y) (Proved)