Simplify sec(x)-tan(x)sin(x)-1/(sec(x))

$\sec (x) - \tan (x) \sin (x) - \frac{1}{\sec (x)}$

Step 1

Simplify terms.

Simplify each term.

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} - \tan (x) \sin (x) - \frac{1}{\sec (x)}$

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

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} \sin (x) - \frac{1}{\sec (x)}$

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

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

$\frac{1}{\cos (x)} - \frac{\sin (x) \sin (x)}{\cos (x)} - \frac{1}{\sec (x)}$

Raise $\sin (x)$ to the power of $1$ .

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

Raise $\sin (x)$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Multiply $\cos (x)$ by $1$ .

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

Combine the numerators over the common denominator.

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

Step 2

Apply pythagorean identity.

$- \cos (x) + \frac{\cos^{2} (x)}{\cos (x)}$

Step 3

Simplify terms.

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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

$- \cos (x) + \frac{\cos (x) \cos (x)}{\cos (x)}$

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

$- \cos (x) + \frac{\cos (x)}{1}$

Divide $\cos (x)$ by $1$ .

$- \cos (x) + \cos (x)$

Add $- \cos (x)$ and $\cos (x)$ .

$0$

Simplify sec(x)-tan(x)sin(x)-1/(sec(x))

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