Trigonometry Examples

$\frac{\tan (t) - \cot (t)}{\sin (t) \cos (t)} = \sec^{2} (t) - \csc^{2} (t)$

Step 1

Start on the right side.

$\sec^{2} (t) - \csc^{2} (t)$

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\sec (t)$ .

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

Step 2.2

Apply the reciprocal identity to $\csc (t)$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify each term.

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

Step 4

Subtract fractions.

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Step 4.1

To write $\frac{1}{\cos^{2} (t)}$ as a fraction with a common denominator, multiply by $\frac{\sin^{2} (t)}{\sin^{2} (t)}$ .

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

Step 4.2

To write $- \frac{1}{\sin^{2} (t)}$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (t)}{\cos^{2} (t)}$ .

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

Step 4.3

Write each expression with a common denominator of $\cos^{2} (t) \sin^{2} (t)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{1}{\cos^{2} (t)}$ by $\frac{\sin^{2} (t)}{\sin^{2} (t)}$ .

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

Step 4.3.2

Multiply $\frac{1}{\sin^{2} (t)}$ by $\frac{\cos^{2} (t)}{\cos^{2} (t)}$ .

$\frac{\sin^{2} (t)}{\cos^{2} (t) \sin^{2} (t)} - \frac{\cos^{2} (t)}{\sin^{2} (t) \cos^{2} (t)}$

Step 4.3.3

Reorder the factors of $\sin^{2} (t) \cos^{2} (t)$ .

$\frac{\sin^{2} (t)}{\cos^{2} (t) \sin^{2} (t)} - \frac{\cos^{2} (t)}{\cos^{2} (t) \sin^{2} (t)}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{\sin^{2} (t) - \cos^{2} (t)}{\cos^{2} (t) \sin^{2} (t)}$

Step 5

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

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

Step 6

Rewrite $\frac{(\sin (t) + \cos (t)) (\sin (t) - \cos (t))}{\cos^{2} (t) \sin^{2} (t)}$ as $\frac{\tan (t) - \cot (t)}{\sin (t) \cos (t)}$ .

$\frac{\tan (t) - \cot (t)}{\sin (t) \cos (t)}$

Step 7

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\tan (t) - \cot (t)}{\sin (t) \cos (t)} = \sec^{2} (t) - \csc^{2} (t)$ is an identity