prove that tan5x+tan3x/tan5x-tan3x=4 cos2x cos4x - Maths - Trigonometric Functions

Question

prove that tan5x+tan3x/tan5x-tan3x=4 cos2x cos4x

Ajanta Trivedi · Accepted Answer

LHS of the given equation is:

$\frac{\tan 5 x + \tan 3 x}{\tan 5 x - \tan 3 x} = \frac{\frac{\sin 5 x}{\cos 5 x} + \frac{\sin 3 x}{\cos 3 x}}{\frac{\sin 5 x}{\cos 5 x} - \frac{\sin 3 x}{\cos 3 x}} = \frac{\sin 5 x \cos 3 x + \sin 3 x \cos 5 x}{\sin 5 x \cos 3 x - \cos 5 x \sin 3 x}$

$= \frac{\sin (5 x + 3 x)}{\sin (5 x - 3 x)} = \frac{\sin 8 x}{\sin 2 x} = \frac{2 \sin 4 x \cos 4 x}{\sin 2 x} [∵ \sin 8 x = 2 \sin 4 x \cos 4 x] = \frac{2 (2 \sin 2 x \cos 2 x) \cos 4 x}{\sin 2 x} = 4 \sin 2 x \cos 4 x$

=RHS

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