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TRIGONOMETRIC FUNCTIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTRIGONOMETRIC FUNCTIONS2 Marks
Question
Prove that : sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
✓

Answer

We have L.H.S. = sin x + sin 3x + sin 5x + sin 7x
= (sin 7x + sin x ) + (sin 5x + sin 3x)
$ = \left[ {2\sin \left( {\frac{{7x + x}}{2}} \right)\cos \left( {\frac{{7x - x}}{2}} \right)} \right]$$ + \left[ {2\sin \left( {\frac{{5x + 3x}}{2}} \right)\cos \left( {\frac{{5x - 3x}}{2}} \right)} \right]$
= 2sin 4xcos3x + 2sin 4xcosx
$\left[ {\therefore \sin \;C + \sin D} \right.$$\left. { = 2\sin \frac{{C + D}}{2} \cdot \cos \frac{{C - D}}{2}} \right]$
= 2 sin 4x [cos 3x + cos x ]
$ = 2\sin 4x\left[ {2\cos \left( {\frac{{3x + x}}{2}} \right)\cos \left( {\frac{{3x - x}}{2}} \right)} \right]$
$\left[ {\because \cos C + \cos D} \right.$$\left. { = 2\cos \frac{{C + D}}{2} \cdot \cos \frac{{C - D}}{2}} \right]$
= 2 sin 4x [2 cos 2x cos x]
= 4cos x cos 2x sin 4x = R.H.S.

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