Precalculus Examples

$\csc (t) (\sin (t) - \tan (t))$

Step 1

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

$\frac{1}{\sin (t)} (\sin (t) - \tan (t))$

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

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Step 6.1

Move the leading negative in $- \frac{1}{\sin (t)}$ into the numerator.

$1 + \frac{- 1}{\sin (t)} \cdot \frac{\sin (t)}{\cos (t)}$

Step 6.2

Cancel the common factor.

$1 + \frac{- 1}{\sin (t)} \cdot \frac{\sin (t)}{\cos (t)}$

Step 6.3

Rewrite the expression.

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

Step 7

Convert from $\frac{1}{\cos (t)}$ to $\sec (t)$ .

$1 - \sec (t)$