Question

If the trigonometric ratios $\sin A$ , $\cos A$ and $\tan A$ are in G.P., then ${{\cos }^{3}}A+{{\cos }^{2}}A$ is equal to
(A) $1$
(B) $2$
(C) $4$
(D) None of these

Answer 1

In this question we have been asked to find the value of ${{\cos }^{3}}A+{{\cos }^{2}}A$ when $\sin A$ , $\cos A$ and $\tan A$ are in geometric progression. We know that when $a,b,c$ are in geometric progression then ${{b}^{2}}=ac$ .

Complete step-by-step solution:
Now considering from the question we have been asked to find the value of ${{\cos }^{3}}A+{{\cos }^{2}}A$ when $\sin A$ , $\cos A$ and $\tan A$ are in geometric progression.
From the basic concepts of progressions we know that when $a,b,c$ are in geometric progression then ${{b}^{2}}=ac$ .
Hence we can say that ${{\cos }^{2}}A=\sin A\left( \tan A \right)$ .
From the concepts of trigonometry we know that $\tan A=\dfrac{\sin A}{\cos A}$ . Hence we can write it as $\Rightarrow {{\cos }^{2}}A=\sin A\left( \dfrac{\sin A}{\cos A} \right)$ . Hence by further simplifying this expression we can say that ${{\cos }^{3}}A={{\sin }^{2}}A$ .
From the basic concepts of trigonometry we know that the trigonometric identity given as ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ is valid.
So we can simplify the expression by using this identity and write it as ${{\cos }^{3}}A=1-{{\cos }^{2}}A$.
We can further simplify this and write it as ${{\cos }^{3}}A+{{\cos }^{2}}A=1$ .
Therefore we can conclude that when $\sin A$ , $\cos A$ and $\tan A$ are in geometric progression then the value of ${{\cos }^{3}}A+{{\cos }^{2}}A$ is one.
Hence we will mark the option “A” as correct.

Note: While answering questions of this type we should be sure with our concepts that we are going to apply and the calculations that we are going to perform in between. If someone had a misconception and wrote it as ${{\cos }^{3}}A={{\cos }^{2}}A$ then we will end up having a mess and cannot answer this question.