Question

How do you use the double angle formula to verify $\sin 4x = 8\cos {x^3} - 4\cos x\sin x?$

Answer 1

Hint: As we know that the above given question is related to trigonometric expression, sine and cosine are trigonometric ratios. Here we have to prove that the left hand side expression is equal to the right hand expression by using the double angle formula. We know that double angle formula which states that $\sin 2a = 2\sin a\cos a$.

Complete step by step answer:
Here we have in the left hand side $\sin 4x$and in the right hand side we have $8{\cos ^3}x - 4\cos x\sin x$.Now by using the identity of $\sin 2a$in left hand side we can write $\sin 4x = \sin (2 \times 2x)$, here $a = 2x$, by substituting the value we get $\sin 4x = 2\sin 2x\cos 2x$. We can further use the same identity in $\sin 2x$, so it can be further written as $2 \times (2\sin x\cos x) \times \cos 2x$.

There is also another double angle formula that we can apply here i.e. $\cos 2x$ can be written as $2{\cos ^2}x - 1$,by applying this in the above equation we get: $\sin 4x = 2 \times (2\sin x\cos ) \times (2{\cos ^2}x - 1)$. By further simplifying we get, $4\sin x\cos x(2{\cos ^2}x - 1) \Rightarrow (4\sin x\cos x) \times (2{\cos ^2}x) - (4\sin x\cos x) \times 1$ . Therefore we have $8{\cos ^3}x\sin x - 4\cos x\sin x$. We can see that this value is equal to the right hand side.

Hence it is verified that $\sin 4x = 8{\cos ^3}x\sin x - 4\cos x\sin x$.

Note: We should note that for the trigonometric ratios we have double angle formula and half angle formula, so by using these formulas, we can solve the trigonometric ratios. The double angle formula for cosine is defined as $\cos (2x) = 2{\cos ^2}x - 1$. Here in the formula $x$ represents the angle. We can also solve this question by using half angle formulas and later we can use double angle formulas or trigonometric identities to solve further.