Question : If $\sin (x+y) = \cos (x–y)$, then the value of $\cos^2 x$ is:
Option 1: $\frac{1}{2}$
Option 2: $3$
Option 3: $5$
Option 4: $\frac{1}{4}$
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Correct Answer: $\frac{1}{2}$
Solution :
Given: $\sin (x+y) = \cos (x–y)$
If $\theta_1+\theta_2=90^{\circ}$, then If $\sin \theta _1=\cos \theta_2$.
$\sin (x+y) = \cos (x–y)$
⇒ $x+y+x–y=90^{\circ}$
⇒ $2x=90^{\circ}$
⇒ $x=45^{\circ}$
The value of $\cos^2 x=\cos^245^{\circ}$
= $(\frac{1}{\sqrt2})^2$
= $\frac{1}{2}$
Hence, the correct answer is $\frac{1}{2}$.
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