Question

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

Answer 1

L.H.S$={\csc }^6\theta$

$={\left({\csc }^2\theta \right)}^3$

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

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

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

$=$R.H.S

Hence proved.