cot(thetha)=3x-1/12x then cot(thetha)+cosec(theta)=6x
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cotθ=3x-1/12x
∴, cot²θ=(3x-1/12x)²
or, cot²θ=9x²-1/2+1/144x²
Now, we knoe that
cosec²θ-cot²θ=1
or, cosec²θ=1+cot²θ
or, cosec²θ=1+9x²-1/2+1/144x²
or, cosec²θ=9x²+1/2+1/144x²
or, cosec²θ=(3x)²+2.3x.1/12x+(1/12x)²
or, cosec²θ=(3x+1/12x)²
or, cosecθ=3x+1/12x
∴, cotθ+cosecθ
=3x-1/12x+3x+1/12x
=6x (Proved)
∴, cot²θ=(3x-1/12x)²
or, cot²θ=9x²-1/2+1/144x²
Now, we knoe that
cosec²θ-cot²θ=1
or, cosec²θ=1+cot²θ
or, cosec²θ=1+9x²-1/2+1/144x²
or, cosec²θ=9x²+1/2+1/144x²
or, cosec²θ=(3x)²+2.3x.1/12x+(1/12x)²
or, cosec²θ=(3x+1/12x)²
or, cosecθ=3x+1/12x
∴, cotθ+cosecθ
=3x-1/12x+3x+1/12x
=6x (Proved)
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