Precalculus Examples

$\cos (- t) + \cos (- t) \tan^{2} (- t)$

Step 1

Multiply by $1$ .

$\cos (- t) \cdot 1 + \cos (- t) \tan^{2} (- t)$

Step 2

Factor $\cos (- t)$ out of $\cos (- t) \tan^{2} (- t)$ .

$\cos (- t) \cdot 1 + \cos (- t) (\tan^{2} (- t))$

Step 3

Factor $\cos (- t)$ out of $\cos (- t) \cdot 1 + \cos (- t) (\tan^{2} (- t))$ .

$\cos (- t) \cdot (1 + \tan^{2} (- t))$

Step 4

Rearrange terms.

$\cos (- t) \cdot (\tan^{2} (- t) + 1)$

Step 5

Apply pythagorean identity.

$\cos (- t) \cdot \sec^{2} (- t)$

Step 6

Since $\cos (- t)$ is an even function, rewrite $\cos (- t)$ as $\cos (t)$ .

$\cos (t) \cdot \sec^{2} (- t)$

Step 7

Since $\sec (- t)$ is an even function, rewrite $\sec (- t)$ as $\sec (t)$ .

$\cos (t) \cdot \sec^{2} (t)$

Step 8

Rewrite $\sec (t)$ in terms of sines and cosines.

$\cos (t) \cdot {(\frac{1}{\cos (t)})}^{2}$

Step 9

Simplify the expression.

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Step 9.1

Apply the product rule to $\frac{1}{\cos (t)}$ .

$\cos (t) \cdot \frac{1^{2}}{\cos^{2} (t)}$

Step 9.2

One to any power is one.

$\cos (t) \cdot \frac{1}{\cos^{2} (t)}$

Step 10

Cancel the common factor of $\cos (t)$ .

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Step 10.1

Factor $\cos (t)$ out of $\cos^{2} (t)$ .

$\cos (t) \cdot \frac{1}{\cos (t) \cos (t)}$

Step 10.2

Cancel the common factor.

$\cos (t) \cdot \frac{1}{\cos (t) \cos (t)}$

Step 10.3

Rewrite the expression.

$\frac{1}{\cos (t)}$

Step 11

Convert from $\frac{1}{\cos (t)}$ to $\sec (t)$ .

$\sec (t)$