Trigonometry Examples

$\cos (θ) \cdot \csc (θ) \cdot \tan (θ) = 1$

Step 1

Start on the left side.

$\cos (θ) \cdot \csc (θ) \cdot \tan (θ)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

Write $\tan (θ)$ in sines and cosines using the quotient identity.

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $\cos (θ)$ .

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Step 3.1.1

Factor $\cos (θ)$ out of $\cos (θ) \cdot \frac{1}{\sin (θ)}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

$\frac{1}{\sin (θ)} \cdot \sin (θ)$

Step 3.2

Combine $\frac{1}{\sin (θ)}$ and $\sin (θ)$ .

$\frac{\sin (θ)}{\sin (θ)}$

Step 3.3

Cancel the common factor of $\sin (θ)$ .

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Step 3.3.1

Cancel the common factor.

$\frac{\sin (θ)}{\sin (θ)}$

Step 3.3.2

Rewrite the expression.

$1$

Step 4

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

$\cos (θ) \cdot \csc (θ) \cdot \tan (θ) = 1$ is an identity