Question : If $\alpha+\beta=90^{\circ}$ and $\alpha=2 \beta$, then the value of $3 \cos ^2 \alpha-2 \sin ^2 \beta$ is equal to:
Option 1: $\frac{3}{4}$
Option 2: $\frac{3}{2}$
Option 3: $\frac{1}{4}$
Option 4: $\frac{4}{3}$
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Correct Answer: $\frac{1}{4}$
Solution :
Given that $\alpha + \beta = 90^\circ$ and $\alpha = 2\beta$,
Substitute $\alpha$ in terms of $\beta$ into the first equation,
$⇒2\beta + \beta = 90^\circ$
$⇒\beta = 30^\circ$
$\therefore \alpha = 60^\circ$
Substitute these values into the expression $3\cos^2\alpha - 2\sin^2\beta$:
$3\cos^2(60^\circ) - 2\sin^2(30^\circ) = 3\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right)^2 = 3\left(\frac{1}{4}\right) - 2\left(\frac{1}{4}\right) = \frac{1}{4}$
Hence, the correct answer is $ \frac{1}{4}$.
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