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$\frac{1+\sin x}{\cos^{2}x}$ =

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Simplify radical expressions involving fractions

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Q.

\frac{1+\sin x}{\cos^{2}x}

=

Apply Pythagorean Identity: Apply the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ to express $\sin^2(x)$ in terms of $\cos^2(x)$ . $\newline$ $\sin^2(x) = 1 - \cos^2(x)$
Rewrite Expression: Rewrite the original expression by adding and subtracting $\sin^2(x)$ in the numerator. $\newline$ $(1 + \sin x)/(\cos^2(x)) = (1 + \sin x + \sin^2(x) - \sin^2(x))/(\cos^2(x))$
Replace with Identity: Use the Pythagorean identity from step $1$ to replace $1 - \sin^2(x)$ with $\cos^2(x)$ . $\newline$ $\frac{1 + \sin x + \sin^2(x) - \sin^2(x)}{\cos^2(x)} = \frac{\cos^2(x) + \sin x}{\cos^2(x)}$
Split Fraction: Split the fraction into two parts. $\newline$ $(\cos^2(x) + \sin x)/(\cos^2(x)) = \cos^2(x)/\cos^2(x) + \sin x/\cos^2(x)$
Simplify First Part: Simplify the first part of the fraction. $\frac{\cos^2(x)}{\cos^2(x)} = 1$
Recognize Trig Identity: Recognize the second part of the fraction as a trigonometric identity. $\sin x/\cos^2(x)$ is the same as $\sin x/(\cos x \cdot \cos x)$ , which is $\tan x/\cos x$ .
Combine Simplified Parts: Combine the simplified parts. $1 + \frac{\tan x}{\cos x}$
Final Simplified Form: Recognize that $\tan x/\cos x$ can be written as $\sin x/\cos^2(x)$ , which is already in the original expression. $\newline$ Therefore, the final simplified form is $1 + \tan x/\cos x$ .

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