Evaluate sec(x)-tan(x)sin(x)=1/(sec(x))

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

Step 1

Simplify $\sec (x) - \tan (x) \sin (x)$ .

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 $1$ .

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

Combine the numerators over the common denominator.

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

Apply pythagorean identity.

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

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

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

$\frac{\cos (x) \cos (x)}{\cos (x)} = \frac{1}{\sec (x)}$

Cancel the common factors.

Multiply by $1$ .

$\frac{\cos (x) \cos (x)}{\cos (x) \cdot 1} = \frac{1}{\sec (x)}$

Cancel the common factor.

$\frac{\cos (x) \cos (x)}{\cos (x) \cdot 1} = \frac{1}{\sec (x)}$

Rewrite the expression.

$\frac{\cos (x)}{1} = \frac{1}{\sec (x)}$

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

$\cos (x) = \frac{1}{\sec (x)}$

Step 2

Simplify $\frac{1}{\sec (x)}$ .

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

$\cos (x) = \frac{1}{\frac{1}{\cos (x)}}$

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

$\cos (x) = 1 \cos (x)$

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

$\cos (x) = \cos (x)$

Step 3

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Step 4

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$x - x = 0$

Subtract $x$ from $x$ .

$0 = 0$

Step 5

Since $0 = 0$ , the equation will always be true.

Always true

Step 6

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$

Evaluate sec(x)-tan(x)sin(x)=1/(sec(x))

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