Question

$\sin A,\cos A,\tan A$ are in GP , then the value of ${\cot }^{6}A-{\cot }^{2}A$ is equal to

• A. $2$
• B. $1$
• C. $0$
• D. $-1$

Answer 1

The correct option is B $1$

Since, $\sin A$ , $\cos A$ and $\tan A$ are in GP
$a=\sin A$
$ar=\cos A$
$a{r}^2=\tan A$
Thus, $r=\frac{\cos A}{\sin A}$
Also, $r=\frac{tanA}{cosA}=\frac{sinA}{{cos}^{2}A}$
Since both the $r$ have to be equal,
${cos}^{3}A={sin}^{2}A$
Or, ${cot}^{3}A=cosecA$
Thus,
$c={cot}^{6}A-{cot}^{2}A={cosec}^{2}A-{cot}^{2}A$
$=1$
Hence, option 'B' is correct.