Verify the identity 2 sin theta 2 cos theta2 4 4 sin 2 th

Verify the identity. (2 sin theta + 2 cos theta)^2 = 4 + 4 sin 2 theta Begin by working with the left side. Square (2 sin theta + 2 cos theta). (Simplify your answer.)

Solution

given

( 2 sin(theta) + 2 cos(theta) )^2 = 4 + 4 sin(2theta)

take

( 2 sin(theta) + 2 cos(theta) )^2

==> ( 2 sin(theta) )^2 + ( 2 cos(theta) )^2 + 2 ( 2 sin(theta) ) ( 2 cos(theta) )

==> 4 sin^2(theta) + 4 cos^2 (theta) + 8( sin(theta) cos(theta) )

==> 4 ( sin^2(theta) + cos^2(theta) ) + 4 ( 2 sin(theta) cos(theta) )

since sin^2(theta) + cos^2(theta) = 1 and sin(2theta) = 2 sin(theta) cos(theta)

so

==> 4 ( 1 ) + 4 ( sin(2theta) )

==> 4 + 4 sin(2theta)