Algebra Examples

$\frac{\tan^{2} (θ) \csc^{2} (θ) - 1}{\sec^{2} (θ)}$

Step 1

Rewrite $\tan^{2} (θ) \csc^{2} (θ)$ as ${(\tan (θ) \csc (θ))}^{2}$ .

$\frac{{(\tan (θ) \csc (θ))}^{2} - 1}{\sec^{2} (θ)}$

Step 2

Rewrite $1$ as $1^{2}$ .

$\frac{{(\tan (θ) \csc (θ))}^{2} - 1^{2}}{\sec^{2} (θ)}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \tan (θ) \csc (θ)$ and $b = 1$ .

$\frac{(\tan (θ) \csc (θ) + 1) (\tan (θ) \csc (θ) - 1)}{\sec^{2} (θ)}$

Step 4

Simplify.

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Step 4.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.1.1

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

$\frac{(\frac{\sin (θ)}{\cos (θ)} \cdot \frac{1}{\sin (θ)} + 1) (\tan (θ) \csc (θ) - 1)}{\sec^{2} (θ)}$

Step 4.1.2

Cancel the common factors.

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

Step 4.2

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

$\frac{(\sec (θ) + 1) (\tan (θ) \csc (θ) - 1)}{\sec^{2} (θ)}$

Step 4.3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factors.

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

Step 4.4

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

$\frac{(\sec (θ) + 1) (\sec (θ) - 1)}{\sec^{2} (θ)}$