Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Prove That: Sin 4 X = 4 Sin X Cos 3 X − 4 Cos X Sin 3 X - Mathematics

Numerical

Prove that: $\sin 4x = 4 \sin x \cos^3 x - 4 \cos x \sin^3 x$

#### Solution

$LHS = sin 4x$

$= 2\sin2x \cos2x \left( \because \sin2\theta = 2sin\theta cos\theta \right)$

Now, using the identities

$\sin2\alpha = 2\sin\alpha\cos\alpha \text{ and } \cos2\alpha = \cos^2 \alpha - \sin^2 \alpha$, we get

$LHS = 2(2\text{ sin } x \text{ cos } x) . ( \cos^2 x - \sin^2 x)$

$= 4\text{ sin } x \cos^3 x - 4 \sin^3 x \text{ cos } x = RHS$

$\text{ Hence proved } .$

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.1 | Q 18 | Page 28