Algebra Examples

\frac{1}{1 + {tan}^{2} (x)} + \frac{1}{1 + {cot}^{2} (x)}

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

Step 1

Rearrange terms.

\frac{1}{{tan}^{2} (x) + 1} + \frac{1}{1 + {cot}^{2} (x)}

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

Step 2

Apply pythagorean identity.

\frac{1}{{sec}^{2} (x)} + \frac{1}{1 + {cot}^{2} (x)}

$\frac{1}{\sec^{2} (x)} + \frac{1}{1 + \cot^{2} (x)}$

Step 3

Apply pythagorean identity.

\frac{1}{{sec}^{2} (x)} + \frac{1}{{csc}^{2} (x)}

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

Step 4

Simplify each term.

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Step 4.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

\frac{1^{2}}{{sec}^{2} (x)} + \frac{1}{{csc}^{2} (x)}

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

Step 4.2

Rewrite

\frac{1^{2}}{{sec}^{2} (x)}

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

{(\frac{1}{sec (x)})}^{2}

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

{(\frac{1}{sec (x)})}^{2} + \frac{1}{{csc}^{2} (x)}

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

Step 4.3

Rewrite

sec (x)

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

{⎛ ⎝ \frac{1}{\frac{1}{cos (x)}} ⎞ ⎠}^{2} + \frac{1}{{csc}^{2} (x)}

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

Step 4.4

Multiply by the reciprocal of the fraction to divide by

\frac{1}{cos (x)}

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

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

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

Step 4.5

Multiply

cos (x)

$\cos (x)$ by

1

$1$ .

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

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

Step 4.6

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 4.7

Rewrite

\frac{1^{2}}{{csc}^{2} (x)}

$\frac{1^{2}}{\csc^{2} (x)}$ as

{(\frac{1}{csc (x)})}^{2}

${(\frac{1}{\csc (x)})}^{2}$ .

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

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

Step 4.8

Rewrite

csc (x)

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

{cos}^{2} (x) + {⎛ ⎝ \frac{1}{\frac{1}{sin (x)}} ⎞ ⎠}^{2}

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

Step 4.9

Multiply by the reciprocal of the fraction to divide by

\frac{1}{sin (x)}

$\frac{1}{\sin (x)}$ .

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

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

Step 4.10

Multiply

sin (x)

$\sin (x)$ by

1

$1$ .

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

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

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

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

Step 5

Rearrange terms.

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

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

Step 6

Apply pythagorean identity.

1

$1$