Prove that: frac{sin 5 x+sin 3 x}{cos 5 x+cos 3 x}=tan 4 x

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3.Trigonometrical Ratios, Functions and Identities

easy

Prove that: $\frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}=\tan 4 x$

Option A

Option B

Option C

Option D

Solution

It is known that

$\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$

$\cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$

$\therefore$ $L.H.S.$ $=\frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}$

$=\frac{2 \sin \left(\frac{5 x+3 x}{2}\right) \cdot \cos \left(\frac{5 x-3 x}{2}\right)}{2 \cos \left(\frac{5 x+3 x}{2}\right) \cdot \cos \left(\frac{5 x-3 x}{2}\right)}$

$=\frac{2 \sin 4 x \cdot \cos x}{2 \cos 4 x \cdot \cos x}$

$=\tan 4 x=R . H . S$

Standard 11

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