Question

If cotθ=3x-1/12x , then show that cotθ+cosecθ=6x or -1/6x

Answer 1

cotθ=3x-1/12x
∴, cot²θ=(3x-1/12x)²
=9x²-2.3x.1/12x+1/144x²
=9x²-1/2+1/144x²
Now, cosec²θ-cot²θ=1
or, cosec²θ=1+cot²θ
or, cosec²θ=1+9x²-1/2+1/144x²
or, cosec²θ=9x²+1/2+1/144x²
or, cosec²θ=9x²+2.3x.1/12x+1/144x²
or, cosec²θ=(3x+1/12x)²
∴, cosecθ=+-(3x+1/12x)
When cosecθ=3x+1/12x,
cotθ+cosecθ
=3x-1/12x+3x+1/12x
=6x
When cosecθ=-(3x+1/12x)
cotθ+cosecθ
=3x-1/12x-3x-1/12x
=-1/6x
∴, cotθ+cosecθ=6x or -1/6x (Proved)