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AP EAMCET · Maths · Trigonometric Ratios & Identities

If \(A+B+C=\frac{\pi}{4}\), then \(\sin 4 A+\sin 4 B+\sin 4 C=\)

  1. A \(4 \cos 2 \mathrm{~A} \cos 2 \mathrm{~B} \cos 2 \mathrm{C}\)
  2. B \(4 \sin 2 \mathrm{~A} \sin 2 \mathrm{~B} \sin 2 \mathrm{C}\)
  3. C \(1+4 \sin 2 \mathrm{~A} \sin 2 \mathrm{~B} \sin 2 \mathrm{C}\)
  4. D \(1+4 \cos 2 \mathrm{~A} \cos 2 \mathrm{~B} \cos 2 \mathrm{C}\)
Verified Solution

Answer & Solution

Correct Answer

(A) \(4 \cos 2 \mathrm{~A} \cos 2 \mathrm{~B} \cos 2 \mathrm{C}\)

Step-by-step Solution

Detailed explanation

Given \(A+B+C=\frac{\pi}{4}\), then \(4A+4B+4C = 4(\frac{\pi}{4}) = \pi\). Using the identity if \(X+Y+Z=\pi\), then \(\sin X + \sin Y + \sin Z = 4 \cos \frac{X}{2} \cos \frac{Y}{2} \cos \frac{Z}{2}\). Let \(X=4A, Y=4B, Z=4C\).…