Chapter 7 - Section 7.1 - Trigonometric Identities - 7.1 Exercises - Page 543: 62

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Prove that $\frac{1 + \sec^2 x}{1 + tan^2 x} = 1 + cos^2 x$ $1 + tan^2 x = sec^2 x $ Thus substitute $sec^2 x$ for $1 + tan^2 x$ $\frac{1 + sec^2 x}{sec^2 x} = 1 + cos^2 x$ $\frac{1}{sec^2 x} = cos^2 x$ Substitute $cos^2 x$ for $\frac{1}{sec^2 x}$ $1 + cos^2 x $ = $1 + cos^2 x$ Hence proved