Trigonometry Examples

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

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

Step 1

Start on the left side.

cos (θ) \cdot csc (θ) \cdot tan (θ)

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

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to

csc (θ)

$\csc (θ)$ .

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

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

Step 2.2

Write

tan (θ)

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

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

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

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of

cos (θ)

$\cos (θ)$ .

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Step 3.1.1

Factor

cos (θ)

$\cos (θ)$ out of

cos (θ) \cdot \frac{1}{sin (θ)}

$\cos (θ) \cdot \frac{1}{\sin (θ)}$ .

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