Question

Express in terms of trig functions of single x : (sin2x+cos2x)^2

Answer 1

$(\sin 2 x+\cos 2 x)^{2}=1+\sin 4 x$

Solution:

Given trigonometric function:

$(\sin 2 x+\cos 2 x)^{2}$

Using perfect square formula:

$(a+b)^{2}=a^{2}+2 a b+b^{2}$

$a=\sin 2 x, b=\cos 2 x$

$(\sin 2 x+\cos 2 x)^{2}=\sin ^{2}(2 x)+2 \sin 2 x \cos 2 x+\cos ^{2}(2 x)$

Arranging square terms together.

                           $=\sin ^{2}(2 x)+\cos ^{2}(2 x)+2 \sin 2 x \cos 2 x$

Using the trigonometric identity: $\sin ^{2}x+\cos ^{2}x=1$

                           $=1+2 \cos 2 x \sin 2 x$

Using the trigonometric identity: $2 \cos \sin x=\sin 2 x$

                          $=1+\sin (2 \cdot 2 x)$

                          $=1+\sin 4 x$

Hence $(\sin 2 x+\cos 2 x)^{2}=1+\sin 4 x$.