Question

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

Answer 1

here is the solution

LHS= tan3x+tan5x/tan5x-tan3x

convert tan into sin and cos,

LHS = (sin5xcos3x+sin3xcos5x)/(sin5xcos3x-sin3xcos5x)

use :sin(a)cos(b)+cos(a)sin(b)=sin(a+b)

and sin(a)cos(b)-sin(b)cos(a)=sin(a-b)

LHS= Sin(8x)/sin(2x)

now use sin(2a)=2sin(a)cos(a)

= 2sin4xcos4x/sin2x

= 4sin2xcos2xcos4x/sin2x

= 4cos2xcos4x= RHS

=> LHS=RHS

hence proved

Answer 2

Answer:

Step-by-step explanation:LHS= tan3x+tan5x/tan5x-tan3x

convert tan into sin and cos,

LHS = (sin5xcos3x+sin3xcos5x)/(sin5xcos3x-sin3xcos5x)

use :sin(a)cos(b)+cos(a)sin(b)=sin(a+b)

and sin(a)cos(b)-sin(b)cos(a)=sin(a-b)

LHS= Sin(8x)/sin(2x)

now use sin(2a)=2sin(a)cos(a)

= 2sin4xcos4x/sin2x

= 4sin2xcos2xcos4x/sin2x= 4cos2xcos4x= RHS

=> LHS=RHS

hence proved