Algebra Examples

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

Step 1

Rearrange terms.

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

Step 2

Apply pythagorean identity.

$\frac{\sec^{2} (θ)}{\cot^{2} (θ)}$

Step 3

Rewrite $\frac{\sec^{2} (θ)}{\cot^{2} (θ)}$ as ${(\frac{\sec (θ)}{\cot (θ)})}^{2}$ .

${(\frac{\sec (θ)}{\cot (θ)})}^{2}$

Step 4

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

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

Step 5

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

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

Step 6

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (θ)}{\sin (θ)}$ .

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

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

${(\sec (θ) \tan (θ))}^{2}$

Step 8

Apply the product rule to $\sec (θ) \tan (θ)$ .

$\sec^{2} (θ) \tan^{2} (θ)$