Precalculus Examples

$\frac{1}{\cos (θ)} + \frac{\tan^{2} (θ)}{\cos (θ)}$

Step 1

Combine the numerators over the common denominator.

$\frac{1 + \tan^{2} (θ)}{\cos (θ)}$

Step 2

Rearrange terms.

$\frac{\tan^{2} (θ) + 1}{\cos (θ)}$

Step 3

Apply pythagorean identity.

$\frac{\sec^{2} (θ)}{\cos (θ)}$

Step 4

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

$\frac{{(\frac{1}{\cos (θ)})}^{2}}{\cos (θ)}$

Step 5

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

$\frac{\frac{1^{2}}{\cos^{2} (θ)}}{\cos (θ)}$

Step 6

One to any power is one.

$\frac{\frac{1}{\cos^{2} (θ)}}{\cos (θ)}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8

Combine.

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

Step 9

Multiply $\cos^{2} (θ)$ by $\cos (θ)$ by adding the exponents.

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Step 9.1

Multiply $\cos^{2} (θ)$ by $\cos (θ)$ .

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Step 9.1.1

Raise $\cos (θ)$ to the power of $1$ .

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

Step 9.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 9.2

Add $2$ and $1$ .

$\frac{1 \cdot 1}{\cos^{3} (θ)}$

Step 10

Multiply $1$ by $1$ .

$\frac{1}{\cos^{3} (θ)}$