Question

prove that : cos7x+cos5x/sin7x+sin5x=cot x​

Answer 1

$\color{red}\huge{\underline{\underline{GIVEN\dag}}}$

• $\frac{ \cos(7x) + \cos(5x) }{ \sin(7x) + \sin(5x) }$

$\color{red}\huge{\underline{\underline{SOLUTION\dag}}}$

We know that;

$\blue{ \boxed{\sin(c) + \sin(d) = 2 \sin( \frac{c + d}{2} ) \cos( \frac{c - d}{2} ) }}$

$\blue{ \boxed{\cos(c) + \cos(d) = 2 \cos( \frac{c + d}{2} ) \cos( \frac{c - d}{2} ) }}$

$\implies \frac{2 \cos( \frac{7x + 5x}{2} ) \cos( \frac{7x - 5x}{2} ) }{2 \sin( \frac{7x + 5x}{2} ) \cos( \frac{7x - 5x}{2} ) }$

$\implies \frac{\cos(6x) \cos(x) }{ \sin(6x) \cos(x) }$

$\orange { \boxed{\therefore \frac{ \sin(7x) + \sin(2x) }{ \cos(7x) + \cos(2x) } = \tan(6x) }}$

HOPE THAT HELPS!!