Trigonometry Examples

$\frac{\cot (A) \csc (A)}{\sec (A)} = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1

Simplify $\frac{\cot (A) \csc (A)}{\sec (A)}$ .

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Step 1.1

Separate fractions.

$\frac{\csc (A)}{1} \cdot \frac{\cot (A)}{\sec (A)} = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.2

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

$\frac{\csc (A)}{1} \cdot \frac{\cot (A)}{\frac{1}{\cos (A)}} = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.3

Rewrite $\cot (A)$ in terms of sines and cosines.

$\frac{\csc (A)}{1} \cdot \frac{\frac{\cos (A)}{\sin (A)}}{\frac{1}{\cos (A)}} = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.4

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

$\frac{\csc (A)}{1} \cdot \frac{\cos (A) \cos (A)}{\sin (A)} = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.5

Convert from $\frac{\cos (A) \cos (A)}{\sin (A)}$ to $\cos (A) \cot (A)$ .

$\frac{\csc (A)}{1} (\cos (A) \cot (A)) = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.6

Divide $\csc (A)$ by $1$ .

$\csc (A) (\cos (A) \cot (A)) = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.7

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

$\frac{1}{\sin (A)} \cos (A) \cot (A) = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.8

Combine $\frac{1}{\sin (A)}$ and $\cos (A)$ .

$\frac{\cos (A)}{\sin (A)} \cot (A) = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.9

Convert from $\frac{\cos (A)}{\sin (A)}$ to $\cot (A)$ .

$\cot (A) \cot (A) = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.10

Multiply $\cot (A) \cot (A)$ .

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Step 1.10.1

Raise $\cot (A)$ to the power of $1$ .

$\cot^{1} (A) \cot (A) = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.10.2

Raise $\cot (A)$ to the power of $1$ .

$\cot^{1} (A) \cot^{1} (A) = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.10.3

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

${\cot (A)}^{1 + 1} = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 1.10.4

Add $1$ and $1$ .

$\cot^{2} (A) = \frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$

Step 2

Simplify $\frac{\cot^{2} (A) + 1}{\tan^{2} (A) + 1}$ .

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Step 2.1

Rearrange terms.

$\cot^{2} (A) = \frac{1 + \cot^{2} (A)}{\tan^{2} (A) + 1}$

Step 2.2

Apply pythagorean identity.

$\cot^{2} (A) = \frac{\csc^{2} (A)}{\tan^{2} (A) + 1}$

Step 2.3

Apply pythagorean identity.

$\cot^{2} (A) = \frac{\csc^{2} (A)}{\sec^{2} (A)}$

Step 2.4

Rewrite $\frac{\csc^{2} (A)}{\sec^{2} (A)}$ as ${(\frac{\csc (A)}{\sec (A)})}^{2}$ .

$\cot^{2} (A) = {(\frac{\csc (A)}{\sec (A)})}^{2}$

Step 2.5

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

$\cot^{2} (A) = {(\frac{\csc (A)}{\frac{1}{\cos (A)}})}^{2}$

Step 2.6

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

$\cot^{2} (A) = {(\frac{\frac{1}{\sin (A)}}{\frac{1}{\cos (A)}})}^{2}$

Step 2.7

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

$\cot^{2} (A) = {(\frac{\cos (A)}{\sin (A)})}^{2}$

Step 2.8

Convert from $\frac{\cos (A)}{\sin (A)}$ to $\cot (A)$ .

$\cot^{2} (A) = \cot^{2} (A)$

Step 3

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

$A = A$

Step 4

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

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Step 4.1

Subtract $A$ from both sides of the equation.

$A - A = 0$

Step 4.2

Subtract $A$ from $A$ .

$0 = 0$

Step 5

Since $0 = 0$ , the equation will always be true for any value of $A$ .

All real numbers

Step 6

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$