Trigonometry Examples

$\frac{\csc^{2} (t)}{\cot (t)} = \csc (t) \sec (t)$

Step 1

Start on the left side.

$\frac{\csc^{2} (t)}{\cot (t)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{{(\frac{1}{\sin (t)})}^{2}}{\cot (t)}$

Step 2.2

Write $\cot (t)$ in sines and cosines using the quotient identity.

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Combine.

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

Step 3.3

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

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Step 3.3.1

Factor $\sin (t)$ out of $1^{2} \sin (t)$ .

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

One to any power is one.

$\frac{1}{\sin (t) \cos (t)}$

Step 4

Reorder terms.

$\frac{1}{\cos (t) \sin (t)}$

Step 5

Rewrite $\frac{1}{\cos (t) \sin (t)}$ as $\csc (t) \sec (t)$ .

$\csc (t) \sec (t)$

Step 6

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

$\frac{\csc^{2} (t)}{\cot (t)} = \csc (t) \sec (t)$ is an identity