Question

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

• A. True
• B. False

Answer 1

The correct option is A True

${\text{csc}}^6\theta -{\cot }^6\theta =({\text{csc}}^2\theta {)}^3-({\cot }^2\theta {)}^3$

$=({\text{csc}}^2\theta -{\cot }^2\theta )({\text{csc}}^4\theta +{\cot }^4\theta +{\text{csc}}^2\theta {\cot }^2\theta )$

$=\left(1\right)\left(\right({\text{csc}}^2\theta {)}^2+({\cot }^2\theta {)}^2-2{\text{csc}}^2\theta {\cot }^2\theta +3{\text{csc}}^2\theta {\cot }^2\theta )$

$=({\text{csc}}^2-{\cot }^2\theta {)}^2+3{\text{csc}}^2\theta {\cot }^2\theta$

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

So the relation is $\text{True}$