Trigonometry Examples

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

Step 1

Square both sides of the equation.

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

Step 2

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

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Simplify each term.

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

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

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

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

Simplify.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

Simplify.

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

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

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

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

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

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

Add $1$ and $1$ .

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

Apply pythagorean identity.

$1^{2} = {(1)}^{2}$

One to any power is one.

$1 = {(1)}^{2}$

Step 3

One to any power is one.

$1 = 1$

Step 4

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$