³≤ft(π/8) ≤ft(3 π/8)+ ³≤ft(π/8) ≤ft(3 π/8)=

Correct Answer is : 122{1}{2 {2}}221

Correct Answer is : 122{1}{2 {2}}221

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MHTCET 11 May 2024 Shift 1 Paper

Question : 143 of 150

Marks: +1, -0

\cos ^{3}\left(\frac{\pi}{8}\right) \cos \left(\frac{3 \pi}{8}\right)+\sin ^{3}\left(\frac{\pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right)=

Solution:

Concept:

Use standard triple-angle identities for

\cos^3 x

and

\sin^3 x

, then simplify using compound angle formulas.

Explanation:

Let

x = \frac{\pi}{8}

, so

3x = \frac{3\pi}{8}

Recall:

\cos^3 x = \frac{1}{4}(\cos 3x + 3\cos x)

\sin^3 x = \frac{1}{4}(3\sin x - \sin 3x)

Rewrite the given expression as

E

E = \cos^3 x \cos 3x + \sin^3 x \sin 3x

E = \frac{1}{4}\left[(\cos 3x + 3\cos x)\cos 3x + (3\sin x - \sin 3x)\sin 3x\right]

Group the terms:

E = \frac{1}{4}\left[(\cos^2 3x - \sin^2 3x) + 3(\cos x\cos 3x + \sin x\sin 3x)\right]

Use

\cos^2 A - \sin^2 A = \cos 2A

and

\cos(A-B) = \cos A\cos B + \sin A\sin B

E = \frac{1}{4}\left[\cos 6x + 3\cos(3x - x)\right]

Substitute

x = \frac{\pi}{8}

6x = \frac{3\pi}{4}

, and

3x - x = 2x = \frac{\pi}{4}

Therefore:

E = \frac{1}{4}\left[\cos\left(\frac{3\pi}{4}\right) + 3\cos\left(\frac{\pi}{4}\right)\right]

= \frac{1}{4}\left[-\frac{1}{\sqrt{2}} + 3\cdot\frac{1}{\sqrt{2}}\right]

= \frac{1}{4}\left(\frac{2}{\sqrt{2}}\right) = \frac{1}{2\sqrt{2}}

Answer:

\frac{1}{2\sqrt{2}}

(Option A)

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