Trigonometry Examples

$\frac{\csc (x)}{\sin (x)} - \frac{\cot (x)}{\tan (x)} = 1$

Step 1

Start on the left side.

$\frac{\csc (x)}{\sin (x)} - \frac{\cot (x)}{\tan (x)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{\frac{1}{\sin (x)}}{\sin (x)} - \frac{\cot (x)}{\tan (x)}$

Step 2.1.2

Rewrite $\frac{\frac{1}{\sin (x)}}{\sin (x)}$ as a product.

$\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)} - \frac{\cot (x)}{\tan (x)}$

Step 2.1.3

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

$\frac{1}{\sin (x) \sin (x)} - \frac{\cot (x)}{\tan (x)}$

Step 2.1.4

Simplify the denominator.

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Step 2.1.4.1

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

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

Step 2.1.4.2

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

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

Step 2.1.4.3

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

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

Step 2.1.4.4

Add $1$ and $1$ .

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

Step 2.1.5

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

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

Step 2.1.6

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

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

Step 2.1.7

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2.1.8

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

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

Step 2.1.9

Simplify the numerator.

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Step 2.1.9.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.9.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.9.3

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

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

Step 2.1.9.4

Add $1$ and $1$ .

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

Step 2.1.10

Simplify the denominator.

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Step 2.1.10.1

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

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Add $1$ and $1$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Apply pythagorean identity.

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

Step 2.4

Cancel the common factor of $\sin^{2} (x)$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$1$

Step 3

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

$\frac{\csc (x)}{\sin (x)} - \frac{\cot (x)}{\tan (x)} = 1$ is an identity