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CUET PG 2023 — Mathematics PYQ

CUET PG | Mathematics | 2023

If sin β is the GM between sin α and cos α, then cos 2β is equal to

Choose the correct answer:

B.
$2 cot^{2} (\frac{π}{4} - α)$

C.
$2 cos^{2} (\frac{π}{4} - α)$

D.
$2 sin^{2} (\frac{π}{4} + α)$

Explanation

Step 1: Geometric Mean Property

Agar $\sin \beta$ , $\sin \alpha$ aur $\cos \alpha$ ka GM hai, toh:

$\sin^2 \beta = \sin \alpha \cdot \cos \alpha$

Dono sides ko $2$ se multiply karne par humein $\sin 2\alpha$ ka formula mil jayega:

$2\sin^2 \beta = 2\sin \alpha \cos \alpha$

$2\sin^2 \beta = \sin 2\alpha$

Step 2: $\cos 2\beta$ Formula

$\cos 2\beta$ ka standard identity use karte hain:

$\cos 2\beta = 1 - 2\sin^2 \beta$

Step 3: Substitution

Ab Step 1 ki value ko Step 2 mein substitute karte hain:

$\cos 2\beta = 1 - \sin 2\alpha$

Alternative Form (Simplification)

Is result ko simplify karne ke liye hum $1$ ko $(\sin^2 \alpha + \cos^2 \alpha)$ likh sakte hain:

$\cos 2\beta = (\sin^2 \alpha + \cos^2 \alpha) - 2\sin \alpha \cos \alpha$

$\cos 2\beta = (\cos \alpha - \sin \alpha)^2$

Final Answer:

$\cos 2\beta = 1 - \sin 2\alpha$