If fx-sin2x-sin2x-3-cosxcosx-3 and g5-4-1- then gofx is equal to

Explanation for the correct option.Given, f(x)=sin^2x+sin^2(x+π/3)+cosxcos(x+π/3) and g(5/4)=1.[ ...

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Standard XII

Mathematics

Evaluation of a Determinant

If fx=sin2x+s...

Question

If $f (x) = \sin^{2} x + \sin^{2} (x + \frac{π}{3}) + \cos x \cos (x + \frac{π}{3})$ and $g (\frac{5}{4}) = 1$ , then $g o f (x)$ is equal to

$1$

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Solution

The correct option is A
$1$

Explanation for the correct option.
Given, $f (x) = \sin^{2} x + \sin^{2} (x + \frac{π}{3}) + \cos x \cos (x + \frac{π}{3})$ and $g (\frac{5}{4}) = 1$ .
$\begin{array}{rcl} f (x) & = & \sin^{2} x + \sin^{2} (x + \frac{π}{3}) + \cos x \cdot \cos (x + \frac{π}{3}) \\ = & \frac{1}{2} [2 \sin^{2} x + 2 \sin^{2} (x + \frac{π}{3}) + 2 \cdot \cos x \cdot \cos (x + \frac{π}{3})] \\ = & \frac{1}{2} [1 - \cos 2 x + 1 - \cos (2 x + \frac{2 π}{3}) + \cos (2 x + \frac{π}{3}) + \cos \frac{π}{3}] [∵ 2 \cdot \cos a \cdot \cos b = \cos (a + b) + \cos (a - b) & & 2 \sin^{2} x = 1 - \cos 2 x] \\ = & \frac{1}{2} [\frac{5}{2} - \cos 2 x - \cos (2 x + \frac{2 π}{3}) + \cos (2 x + \frac{π}{3})] \\ = & \frac{1}{2} [\frac{5}{2} - 2 \cos (2 x + \frac{π}{3}) \cos \frac{π}{3} + \cos (2 x + \frac{π}{3})] \\ = & \frac{1}{2} [\frac{5}{2} - \cos (2 x + \frac{π}{3}) + \cos (2 x + \frac{π}{3})] \\ = & \frac{5}{4} \end{array}$
$\Rightarrow f (x) = \frac{5}{4}$
Now, find $g o f (x)$ .
$\begin{array}{rcl} g o f (x) & = & g (f (x)) \\ = & g (\frac{5}{4}) \\ = & 1 [Given : g (\frac{5}{4}) = 1] \end{array}$
Hence, the correct option is A.

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If f(x)=sin2x+sin2x+π3+cosxcosx+π3 and g54=1, then gof(x) is equal to

If $f (x) = \sin^{2} x + \sin^{2} (x + \frac{π}{3}) + \cos x \cos (x + \frac{π}{3})$ and $g (\frac{5}{4}) = 1$ , then $g o f (x)$ is equal to