Question

Prove that:
${\csc }^6\theta ={\cot }^6\theta +3{\cot }^2\theta {\csc }^2\theta +1$

Answer 1

To prove:- ${\csc }^6\theta ={\cot }^6\theta +3{\cot }^2\theta {\csc }^2\theta +1$

Proof:-

Taking L.H.S., we have

${\csc }^6\theta$

$={(1+{\cot }^2\theta )}^3(\because 1+{\cot }^2\theta ={\csc }^2\theta )$

$=1+{\cot }^6\theta +3{\cot }^2\theta (1+{\cot }^2\theta )(\because {(x+y)}^3=x^3+y^3+3xy(x+y))$

$={\cot }^6\theta +3{\cot }^2\theta {\csc }^2\theta +1$

$=$ R.H.S.

Hence proved.