Trigonometry Examples

$\csc^{2} (y) - \csc (y) \cot (y) = \frac{1}{1 + \cos (y)}$

Step 1

Start on the left side.

$\csc^{2} (y) - \csc (y) \cot (y)$

Step 2

Simplify each term.

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Step 2.1

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

${(\frac{1}{\sin (y)})}^{2} - \csc (y) \cot (y)$

Step 2.2

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

$\frac{1^{2}}{\sin^{2} (y)} - \csc (y) \cot (y)$

Step 2.3

One to any power is one.

$\frac{1}{\sin^{2} (y)} - \csc (y) \cot (y)$

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $- \frac{1}{\sin (y)} \cdot \frac{\cos (y)}{\sin (y)}$ .

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Step 2.6.1

Multiply $\frac{\cos (y)}{\sin (y)}$ by $\frac{1}{\sin (y)}$ .

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

Step 2.6.2

Raise $\sin (y)$ to the power of $1$ .

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

Step 2.6.3

Raise $\sin (y)$ to the power of $1$ .

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

Step 2.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.6.5

Add $1$ and $1$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Simplify.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

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

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

Step 5.1.2

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

$\frac{1 - \cos (y)}{(1 + \cos (y)) (1 - \cos (y))}$

Step 5.2

Cancel the common factor of $1 - \cos (y)$ .

$\frac{1}{1 + \cos (y)}$

Step 6

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

$\csc^{2} (y) - \csc (y) \cot (y) = \frac{1}{1 + \cos (y)}$ is an identity