Question

How do you prove $\sin 3x = 3{\cos ^2}x\sin x - {\sin ^3}x$?

Answer 1

Hint:In order to proof the above statement ,first take the left hand side of the equation and write $3x$ as $2x + x$ and apply the formula of sum of angles of sine as $\sin (A + B) = \sin A\cos B + \cos A\sin B$.Now use the formula of $\sin 2x = 2\sin x\cos x$ and $\cos 2x = {\cos ^2}x - {\sin ^2}x$ to simplify the left-hand side and combine all the like terms, you will get left-hand side equal to right hand side .

Complete step by step solution:
To prove: $\sin 3x = 3{\cos ^2}x\sin x - {\sin ^3}x$
Proof: In order to prove the above equation, we will be first taking the left-hand side of the equation and do some operations
Taking Left-hand Side of the equation,
$ \Rightarrow \sin 3x$
writing $3x$ as $2x + x$,we get
$ \Rightarrow \sin \left( {2x + x} \right)$
Now applying the sum of angle formula of sine $\sin (A + B) = \sin A\cos B + \cos A\sin B$, we get
$ \Rightarrow \sin 2x\cos x + \cos 2x\sin x$
As we know that $\sin 2x = 2\sin x\cos x$and $\cos 2x = {\cos ^2}x - {\sin ^2}x$,putting these values in above
$ \Rightarrow \left( {2\sin x\cos x} \right)\cos x + \left( {{{\cos }^2}x - {{\sin }^2}x} \right)\sin x$
Simplifying further,
$ \Rightarrow 2\sin x{\cos ^2}x + \sin x{\cos ^2}x - {\sin ^3}x$
Combining like terms,
\[ \Rightarrow 3\sin x{\cos ^2}x - {\sin ^3}x\]
$\therefore LHS = 3\sin x{\cos ^2}x - {\sin ^3}x$
Taking Right-hand Side part of the equation
$RHS = 3\sin x{\cos ^2}x - {\sin ^3}x$
$\therefore LHS = RHS$
Hence, proved.

Additional Information:
1. Trigonometry is one of the significant branches throughout the entire existence of mathematics and this idea is given by a Greek mathematician Hipparchus.
2. Even Function – A function $f(x)$ is said to be an even function ,if $f( - x) = f(x)$for all x in its domain.
Odd Function – A function $f(x)$ is said to be an even function ,if $f( - x) = - f(x)$for all x in its domain.
We know that $\sin ( - \theta ) = - \sin \theta .\cos ( - \theta ) = \cos \theta \,and\,\tan ( - \theta )
= - \tan \theta $
Therefore, $\sin \theta $ and $\tan \theta $ and their reciprocals, $\cosec\theta $ and $\cot \theta $ are odd functions whereas \[\cos \theta \] and its reciprocal \[\sec \theta \] are even functions.

Note:
1.One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer.
2.Fromula should be correctly used at every point.