Trigonometry Examples

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

Step 1

Start on the left side.

$\frac{\cos (x)}{\sec (x)} + \frac{\sin (x)}{\csc (x)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{\cos (x)}{\frac{1}{\cos (x)}} + \frac{\sin (x)}{\csc (x)}$

Step 2.1.2

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

$\cos (x) \cos (x) + \frac{\sin (x)}{\csc (x)}$

Step 2.1.3

Simplify.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Add $1$ and $1$ .

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.1.6

Simplify.

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Add $1$ and $1$ .

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

Step 2.2

Rearrange terms.

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

Step 2.3

Apply pythagorean identity.

$1$

Step 3

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

$\frac{\cos (x)}{\sec (x)} + \frac{\sin (x)}{\csc (x)} = 1$ is an identity