Answer
See the explanation below.
Work Step by Step
${{\tan }^{2}}2x+{{\sin }^{2}}2x+{{\cos }^{2}}2x$
Applying the Pythagorean identity of trigonometry ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ , then the above expression can be further simplified as:
${{\tan }^{2}}2x+{{\sin }^{2}}2x+{{\cos }^{2}}2x={{\tan }^{2}}2x+1$
By using the Pythagorean identity of trigonometry ${{\sec }^{2}}x=1+{{\tan }^{2}}x$ , the above expression can be further simplified as:
${{\tan }^{2}}2x+1={{\sec }^{2}}2x$
Thus, the left side of the expression is equal to the right side, which is
${{\tan }^{2}}2x+{{\sin }^{2}}2x+{{\cos }^{2}}2x={{\sec }^{2}}2x$.
