What is ²≤ft(π/4+θ)-²≤ft(π/4-θ) equal to?

Correct Answer is : sin⁡2θ 2θsin2θ

Correct Answer is : sin⁡2θ 2θsin2θ

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UPSC NDA I 2022 Mathematics Solved Paper

Question : 33 of 120

Marks: +1, -0

What is

\sin^2\left(\frac{\pi}{4}+\theta\right)-\sin^2\left(\frac{\pi}{4}-\theta\right)

equal to?

Solution:

Concept:

Use complementary angle and double-angle identities to simplify the expression.

Explanation:

Let

y = \sin^2\left(\frac{\pi}{4}+\theta\right) - \sin^2\left(\frac{\pi}{4}-\theta\right)

First, convert the first sine term to a cosine using

\sin \alpha = \cos\left(\frac{\pi}{2} - \alpha\right)

Set

\alpha = \frac{\pi}{4}+\theta

, then

\sin\left(\frac{\pi}{4}+\theta\right) = \cos\left(\frac{\pi}{2} - \left(\frac{\pi}{4}+\theta\right)\right) = \cos\left(\frac{\pi}{4}-\theta\right)

Thus

\sin^2\left(\frac{\pi}{4}+\theta\right) = \cos^2\left(\frac{\pi}{4}-\theta\right)

Now

y = \cos^2\left(\frac{\pi}{4}-\theta\right) - \sin^2\left(\frac{\pi}{4}-\theta\right)

Apply the identity

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

with

\beta = \frac{\pi}{4} - \theta

Therefore

y = \cos\left(2\left(\frac{\pi}{4}-\theta\right)\right) = \cos\left(\frac{\pi}{2} - 2\theta\right)

Finally,

\cos\left(\frac{\pi}{2} - 2\theta\right) = \sin 2\theta

y = \sin 2\theta

Answer:

\sin 2\theta

(Option A).

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