Precalculus Examples

$\cot (θ) \tan^{3} (θ) + 1$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{\cos (θ)}{\sin (θ)} \tan^{3} (θ) + 1$

Step 1.2

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

$\frac{\cos (θ)}{\sin (θ)} {(\frac{\sin (θ)}{\cos (θ)})}^{3} + 1$

Step 1.3

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

$\frac{\cos (θ)}{\sin (θ)} \cdot \frac{\sin^{3} (θ)}{\cos^{3} (θ)} + 1$

Step 1.4

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

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Step 1.4.1

Factor $\cos (θ)$ out of $\cos^{3} (θ)$ .

$\frac{\cos (θ)}{\sin (θ)} \cdot \frac{\sin^{3} (θ)}{\cos (θ) \cos^{2} (θ)} + 1$

Step 1.4.2

Cancel the common factor.

$\frac{\cos (θ)}{\sin (θ)} \cdot \frac{\sin^{3} (θ)}{\cos (θ) \cos^{2} (θ)} + 1$

Step 1.4.3

Rewrite the expression.

$\frac{1}{\sin (θ)} \cdot \frac{\sin^{3} (θ)}{\cos^{2} (θ)} + 1$

Step 1.5

Cancel the common factor of $\sin (θ)$ .

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Step 1.5.1

Factor $\sin (θ)$ out of $\sin^{3} (θ)$ .

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

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

Step 1.6

Convert from $\frac{\sin^{2} (θ)}{\cos^{2} (θ)}$ to $\tan^{2} (θ)$ .

$\tan^{2} (θ) + 1$

Step 2

Apply pythagorean identity.

$\sec^{2} (θ)$