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Chapter 5: Problem 48

Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\frac{\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x}$$

Short Answer

Expert verified
The simplified form of the given expression is \(1 + 2\sin x\).

Step by step solution

01

Distribute the Denominators

Distribute the denominators across the fractions: \((\cos x)(\cos x) + (1+\sin x)(1+\sin x)\).
02

Simplify the Expression

The expression simplifies to \(\cos^2 x + 1 + 2\sin x + sin^2 x\). Now, replace \(1 + \sin^2 x\) with \(\cos^2 x\), according to Pythagorean identity.
03

Replace and Rearrange Terms

Replace \(\cos^2 x\) by \(1 - \sin^2 x\). Therefore, we have \(1 - \sin^2 x + 2\sin x + \sin^2 x\). The terms \(- \sin^2 x + \sin^2 x\) cancel each other.
04

Final Simplification

Thus, the expression simplifies to \(1 + 2\sin x\).

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