# Chapter 5 - Section 5.1 - Verifying Trigonometric Identities - Exercise Set - Page 659: 39

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 ${{\tan }^{2}}2x+{{\sin }^{2}}2x+{{\cos }^{2}}2x={{\sec }^{2}}2x$.

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