Simplify the trigonometric expression sin4x 2an2x using Dou

Simplify the trigonometric expression sin(4x) + 2an(2x) using Double-Angle Identities. 8sin(x)cos^3(x) 8sin(x)cos^3(x) - 8sin(x)cos(x) 8sin(x)cos^3(x) - 4sin^3(x)cos(x) 8sin(x)cos^3(x)+8sin(x)cos(x)

Solution

sin(4x) +2sin(2x)

sin(2a) =2sin(a)cos(a)

=2sin(2x)cos(2x) +2sin(2x)

=2sin(2x) [cos(2x)  +1]

sin(2a) =2sin(a)cos(a),cos(2a) =cos²(a)-sin²(a) ,sin²(a) +cos²(a) =1

=2(2sin(x)cos(x)) [cos²(x)-sin²(x)  +sin²(x) +cos²(x)]

=4sin(x)cos(x) [2cos²(x)]

=8sin(x)cos³(x)

sin(4x) +2sin(2x)=8sin(x)cos³(x)