15. cot 4x (sin 5x +sin 3x) cot x (sin 5x - sin 3x)
The given trigonometric equation is,
cot 4 x ( sin 5 x + sin 3 x ) = cot x ( sin 5 x − sin 3 x )
Solve left hand side,
L .H .S . = cot 4 x ( sin 5 x + sin 3 x ) = cot 4 x × 2 sin ( 5 x + 3 x ) 2 cos ( 5 x − 3 x ) 2 = cot 4 x × 2 sin 8 x 2 cos 2 x 2 = cos 4 x sin 4 x × 2 sin 4 x cos x
Further simplify,
L .H .S . = 2 cos 4 x cos x = 2 cos 4 x sin x × cos x sin x = [ { sin ( 4 x + x ) − sin ( 4 x − x ) } ] × cot x = cot x ( sin 5 x − sin 3 x )
Right hand side is,
R .H .S . = cot x ( sin 5 x − sin 3 x )
Hence, L .H .S . = R .H .S .
The given trigonometric equation is,
Solve left hand side,
Further simplify,
Right hand side is,
Hence,
