Question

prove that cos C + cos D = 2cos C+D/2 cos C-D/2​

Answer 1

$\sf \purple{we \: know \: that}$

$\red{ \boxed{\cos x = \cos^{2} \frac{x}{2} - \sin^{2} \frac{x}{2} }}$

$\implies \sf \cos^{2} \frac{C}{2} - \sin^{2} \frac{C}{2} + \cos^{2} \frac{D}{2} - \sin^{2} \frac{D}{2}$

$\implies \sf \cos^{2} \frac{C}{2} - (1 - \cos^{2} \frac{C}{2}) + \cos^{2} \frac{D}{2} - (1 - \cos^{2} \frac{D}{2})$

$\implies \sf 2\cos^{2} \frac{C}{2} + 2\cos^{2} \frac{D}{2} -2$

$\implies \sf 2(\cos^{2} \frac{C}{2} + \cos^{2} \frac{D}{2} -1)$

$\implies \sf 2(\cos^{2} \frac{C}{2} - ( 1 - \cos^{2} \frac{D}{2} ))$

$\implies \sf 2(\cos^{2} \frac{C}{2} - \sin^{2} \frac{D}{2} )$

$\blue{ \boxed{\cos^{2} a - \sin^{2} b = \cos(a + b) \cos(a - b) }}$

$\orange{\therefore \sf 2 \: \: cos \small \:{\frac{C+D}{2} } \: \: cos \: \frac{C - D}{2} }$

HOPE IT HELPS !