Trigonometry Examples

$\csc (θ) - \cos (θ) \cot (θ) = \sin (θ)$

Step 1

Start on the left side.

$\csc (θ) - \cos (θ) \cot (θ)$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{1}{\sin (θ)} - \cos (θ) \cot (θ)$

Step 2.1.2

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

$\frac{1}{\sin (θ)} - \cos (θ) \frac{\cos (θ)}{\sin (θ)}$

Step 2.1.3

Multiply $- \cos (θ) \frac{\cos (θ)}{\sin (θ)}$ .

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Step 2.1.3.1

Combine $\frac{\cos (θ)}{\sin (θ)}$ and $\cos (θ)$ .

$\frac{1}{\sin (θ)} - \frac{\cos (θ) \cos (θ)}{\sin (θ)}$

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

Add $1$ and $1$ .

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

Step 2.2

Combine the numerators over the common denominator.

$\frac{1 - \cos^{2} (θ)}{\sin (θ)}$

Step 2.3

Apply pythagorean identity.

$\frac{\sin^{2} (θ)}{\sin (θ)}$

Step 2.4

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

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Step 2.4.1

Factor $\sin (θ)$ out of $\sin^{2} (θ)$ .

$\frac{\sin (θ) \sin (θ)}{\sin (θ)}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Multiply by $1$ .

$\frac{\sin (θ) \sin (θ)}{\sin (θ) \cdot 1}$

Step 2.4.2.2

Cancel the common factor.

$\frac{\sin (θ) \sin (θ)}{\sin (θ) \cdot 1}$

Step 2.4.2.3

Rewrite the expression.

$\frac{\sin (θ)}{1}$

Step 2.4.2.4

Divide $\sin (θ)$ by $1$ .

$\sin (θ)$

Step 3

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

$\csc (θ) - \cos (θ) \cot (θ) = \sin (θ)$ is an identity