Question

If sin $\theta$ , cos $\theta$ , tan $\theta$ are in G.P. then $co{s}^9\theta +co{s}^6\theta +3co{s}^5\theta -1$ is equal to

• A. -1
• B. 0
• C. 1
• D. 2

Answer 1

Answer: (B)

Given : sin $\theta$ , cos $\theta$ , tan $\theta$ are in G.P.
$\Rightarrow co{s}^2\theta =sin\theta tan\theta \Rightarrow co{s}^3\theta =si{n}^2\theta$
$\Rightarrow co{s}^3\theta =1-co{s}^2\theta$
$\Rightarrow (co{s}^3\theta +co{s}^2\theta )=1...(1)$
Cubic both sides, we have
$co{s}^9\theta +co{s}^6\theta +3co{s}^5\theta .(co{s}^3\theta +co{s}^2\theta )=1$
$\Rightarrow co{s}^9\theta +co{s}^6\theta +3co{s}^5\theta =1$
[Using equation (1)]
$\Rightarrow co{s}^9\theta +co{s}^6\theta +3co{s}^5\theta -1=0$