Q. 23.9( 47 Votes )

Prove that (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0

Answer :

To Prove: (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0

Formulas to use:




Proof:

Replacing x with 3x and y with x in the formula, we get-



sin 3x + sin x = 2 sin 2x cos x …(1)


Similarly,


We know that



Replacing x with 3x and y with x, we get-



cos 3x - cos x = -2 sin 2x sin x …(2)


Now,


L.H.S = (sin 3x + sin x) sin x + (cos 3x – cos x) cos x

Putting the values obtained from equations 1 and 2 we get,

= (2 sin 2x cos x) sin x + (-2 sin 2x sin x) cos x

= 2 sin 2x cos x sin x - 2 sin 2x sin x cos x

= 0


= R.H.S


Hence, L.H.S = R.H.S…Proved

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