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Question

# If $f\left(x\right)={\mathrm{sin}}^{2}x+{\mathrm{sin}}^{2}\left(x+\frac{\mathrm{\pi }}{3}\right)+\mathrm{cos}x\mathrm{cos}\left(x+\frac{\mathrm{\pi }}{3}\right)$ and $g\left(\frac{5}{4}\right)=1$, then $gof\left(x\right)$ is equal to

A

$1$

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B

$-1$

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C

$2$

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D

$-2$

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Solution

## The correct option is A $1$Explanation for the correct option.Given, $f\left(x\right)={\mathrm{sin}}^{2}x+{\mathrm{sin}}^{2}\left(x+\frac{\mathrm{\pi }}{3}\right)+\mathrm{cos}x\mathrm{cos}\left(x+\frac{\mathrm{\pi }}{3}\right)$ and $g\left(\frac{5}{4}\right)=1$.$\begin{array}{rcl}f\left(x\right)& =& {\mathrm{sin}}^{2}x+{\mathrm{sin}}^{2}\left(x+\frac{\mathrm{\pi }}{3}\right)+\mathrm{cos}x·\mathrm{cos}\left(x+\frac{\mathrm{\pi }}{3}\right)\\ & =& \frac{1}{2}\left[2{\mathrm{sin}}^{2}x+2{\mathrm{sin}}^{2}\left(x+\frac{\mathrm{\pi }}{3}\right)+2·\mathrm{cos}x·\mathrm{cos}\left(x+\frac{\mathrm{\pi }}{3}\right)\right]\\ & =& \frac{1}{2}\left[1-\mathrm{cos}2x+1-\mathrm{cos}\left(2x+\frac{2\mathrm{\pi }}{3}\right)+\mathrm{cos}\left(2x+\frac{\mathrm{\pi }}{3}\right)+\mathrm{cos}\frac{\mathrm{\pi }}{3}\right]\mathbf{\left[}\mathbf{\because }\mathbf{2}\mathbf{·}\mathbf{cos}\mathbf{a}\mathbf{·}\mathbf{cos}\mathbf{b}\mathbf{=}\mathbf{cos}\mathbf{\left(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{\right)}\mathbf{+}\mathbf{cos}\mathbf{\left(}\mathbf{a}\mathbf{-}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{&}\mathbf{&}\mathbf{}\mathbf{2}{\mathbf{sin}}^{\mathbf{2}}\mathbf{x}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{cos}\mathbf{2}\mathbf{x}\mathbf{\right]}\\ & =& \frac{1}{2}\left[\frac{5}{2}-\mathrm{cos}2x-\mathrm{cos}\left(2x+\frac{2\mathrm{\pi }}{3}\right)+\mathrm{cos}\left(2x+\frac{\mathrm{\pi }}{3}\right)\right]\\ & =& \frac{1}{2}\left[\frac{5}{2}-2\mathrm{cos}\left(2x+\frac{\mathrm{\pi }}{3}\right)\mathrm{cos}\frac{\mathrm{\pi }}{3}+\mathrm{cos}\left(2x+\frac{\mathrm{\pi }}{3}\right)\right]\\ & =& \frac{1}{2}\left[\frac{5}{2}-\mathrm{cos}\left(2x+\frac{\mathrm{\pi }}{3}\right)+\mathrm{cos}\left(2x+\frac{\mathrm{\pi }}{3}\right)\right]\\ & =& \frac{5}{4}\end{array}$$⇒f\left(x\right)=\frac{5}{4}$Now, find $gof\left(x\right)$.$\begin{array}{rcl}gof\left(x\right)& =& g\left(f\left(x\right)\right)\\ & =& g\left(\frac{5}{4}\right)\\ & =& 1\mathbf{}\mathbf{\left[}\mathbf{Given}\mathbf{:}\mathbf{}\mathbf{g}\left(\frac{5}{4}\right)\mathbf{=}\mathbf{1}\mathbf{\right]}\end{array}$Hence, the correct option is A.

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