Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Now consider the right side of the equation.

$\csc (t) - \sin (t)$

Step 4

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

$\frac{1}{\sin (t)} - \sin (t)$

Step 5

Write $- \sin (t)$ as a fraction with denominator $1$ .

$\frac{1}{\sin (t)} + \frac{- \sin (t)}{1}$

Step 6

Add fractions.

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Step 6.1

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

$\frac{1}{\sin (t)} + \frac{- \sin (t)}{1} \cdot \frac{\sin (t)}{\sin (t)}$

Step 6.2

Multiply $\frac{- \sin (t)}{1}$ by $\frac{\sin (t)}{\sin (t)}$ .

$\frac{1}{\sin (t)} + \frac{- \sin (t) \sin (t)}{\sin (t)}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{1 - \sin (t) \sin (t)}{\sin (t)}$

Step 7

Multiply $- \sin (t) \sin (t)$ .

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

Step 8

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

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