Question

If cosec theta = x + 1/4x prove that cosec theta + cot theta = 2x or 1/2x

Answer 1

cosecθ=x+1/4x
∴, cosec²θ=(x+1/4x)²
=x²+2.x.1/4x+1/16x²
=x²+1/2+1/16x²
Now, cosec²θ-cot²θ=1
or, cot²θ=cosec²θ-1
or, cot²θ=x²+1/2+1/16x²-1
or, cot²θ=x²+1/16x²-1/2
or, cot²θ=x²-2.x.1/4x+1/16x²
or, cot²θ=(x-1/4x)²
or, cotθ=+-(x-1/4x)
∴, either, cosecθ+tanθ
=x+1/4x+x-1/4x [when cotθ=x+1/4x]
=2x  
or, cosecθ+cotθ
=x+1/4x-x+1/4x [when cotθ=-(x+1/4x)]
=1/4x+1/4x
=2/4x
=1/2x (Proved)

Answer 2

Answer: Please find an attachment image

Step-by-step explanation:

Hope it helps...