Prove the following identity:
$(s i n 3 x + s i n x) s i n x + (c o s 3 x - c o s x) c o s x = 0$

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Solution

To prove:

$(s i n 3 x + s i n x) s i n x + (c o s 3 x - c o s x) c o s x = 0$

L.H.S. $= (s i n 3 x + s i n x) s i n x + (c o s 3 x - c o s x) c o s x$

$⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} ∵ s i n C + s i n D = 2 s i n \frac{C + D}{2} c o s \frac{C - D}{2} c o s C - c o s D = - 2 s i n \frac{C + D}{2} s i n \frac{C - D}{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

$= 2 s i n 2 x c o s x s i n x + (- 2 s i n 2 x . s i n x c o s x)$

$= 2 s i n 2 x c o s x s i n x - 2 s i n 2 x c o s x s i n x)$

$= 0$

$= R . H . S .$

Hence, proved.

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