Question

If $\sin \theta$ , $\cos \theta$ and $\tan \theta$ are in G.P. then ${\cot }^6\theta -{\cot }^2\theta$ is

• A. $1$
• B. $1/2$
• C. $2$
• D. $3$

Answer 1

The correct option is A $1$

As they are in GP

So, $\tan \theta \sin \theta ={\cos }^2\theta \Rightarrow {\sin }^2\theta ={\cos }^3\theta$

Now, ${\cot }^6\theta =\frac{{\cos }^6\theta }{{\sin }^6\theta }=\frac{{\cos }^6\theta }{({\sin }^2\theta {)}^3}=\frac{{\cos }^6\theta }{({\cos }^3\theta {)}^3}=\frac{1}{{\cos }^3\theta }=\frac{1}{{\sin }^2\theta }={\text{cosec}}^2\theta$

Hence, ${\cot }^6\theta -{\cot }^2\theta ={\text{cosec}}^2\theta -{\cot }^2\theta =1$

Therefore, Answer is $1$