Trigonometry Examples

$2 - \csc^{2} (x) = 1 - \cot^{2} (x)$

Step 1

Start on the right side.

$1 - \cot^{2} (x)$

Step 2

Apply Pythagorean identity in reverse.

$1 - (\csc^{2} (x) - 1)$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Add $1$ and $1$ .

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

Step 5

Reorder terms.

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

Step 6

Rewrite $2 - \frac{1}{\sin^{2} (x)}$ as $2 - \csc^{2} (x)$ .

$2 - \csc^{2} (x)$

Step 7

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

$2 - \csc^{2} (x) = 1 - \cot^{2} (x)$ is an identity