Prove the identity: (sin 4x)/(sin x)=4cos x cos 2x

Apply double angle formula: Use the double angle formula for sine to express sin 4x in terms of sin 2x and cos 2x. The double angle formula for sine is sin 2θ = 2sin θ cos θ. We can apply this formula to sin 4x by setting θ = 2x, which gives us sin 4x = 2sin 2x cos 2x.Apply formula to sin 2x: Apply the double angle formula again to sin 2x. Using the same double angle formula, sin 2θ = 2sin θ cos θ, we can express sin 2x as 2sin x cos x. So, sin 4x becomes 2(2sin x cos x)cos 2x.Substitute into original identity: Substitute the expression from Step 2 into the original identity. We replace sin 4x in the original identity (sin 4x)/(sin x) with 2(2sin x cos x)cos 2x to get (2(2sin x cos x)cos 2x)/(sin x).Cancel out sin x: Simplify the expression by canceling out sin x. The sin x in the numerator and denominator cancel each other out, leaving us with 2(2cos x cos 2x).Multiply constants: Simplify the expression further by multiplying the constants. Multiplying the constants 2 and 2 gives us 4cos x cos 2x, which is the right-hand side of the original identity.

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Q. Prove the identity:

\newline

\frac{\sin 4x}{\sin x} = 4\cos x \cos 2x

Apply double angle formula: Use the double angle formula for sine to express $\sin 4x$ in terms of $\sin 2x$ and $\cos 2x$ . The double angle formula for sine is $\sin 2\theta = 2\sin \theta \cos \theta$ . We can apply this formula to $\sin 4x$ by setting $\theta = 2x$ , which gives us $\sin 4x = 2\sin 2x \cos 2x$ .
Apply formula to $\sin 2x$ : Apply the double angle formula again to $\sin 2x$ . Using the same double angle formula, $\sin 2\theta = 2\sin \theta \cos \theta$ , we can express $\sin 2x$ as $2\sin x \cos x$ . So, $\sin 4x$ becomes $2(2\sin x \cos x)\cos 2x$ .
Substitute into original identity: Substitute the expression from Step $2$ into the original identity. $\newline$ We replace $\sin 4x$ in the original identity $\frac{\sin 4x}{\sin x}$ with $2(2\sin x \cos x)\cos 2x$ to get $\frac{2(2\sin x \cos x)\cos 2x}{\sin x}$ .
Cancel out $\sin x$ : Simplify the expression by canceling out $\sin x$ . The $\sin x$ in the numerator and denominator cancel each other out, leaving us with $2(2\cos x \cos 2x)$ .
Multiply constants: Simplify the expression further by multiplying the constants. Multiplying the constants $2$ and $2$ gives us $4\cos x \cos 2x$ , which is the right-hand side of the original identity.