Trigonometry Examples

1 - \frac{1}{{cos}^{2} (x)}

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

1 - \frac{1^{2}}{{cos}^{2} (x)}

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

Step 1.1.2

Rewrite

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

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

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

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

1 - {(\frac{1}{cos (x)})}^{2}

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

Step 1.1.3

Convert from

\frac{1}{cos (x)}

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

sec (x)

$\sec (x)$ .

1 - {sec}^{2} (x)

$1 - \sec^{2} (x)$

1 - {sec}^{2} (x)

$1 - \sec^{2} (x)$

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Reorder

1

$1$ and

- {sec}^{2} (x)

$- \sec^{2} (x)$ .

- {sec}^{2} (x) + 1

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

Step 1.2.2

Factor

- 1

$- 1$ out of

- {sec}^{2} (x)

$- \sec^{2} (x)$ .

- ({sec}^{2} (x)) + 1

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

Step 1.2.3

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

- {sec}^{2} (x) - 1 \cdot - 1

$- \sec^{2} (x) - 1 \cdot - 1$

Step 1.2.4

Factor

- 1

$- 1$ out of

- {sec}^{2} (x) - 1 \cdot - 1

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

- ({sec}^{2} (x) - 1)

$- (\sec^{2} (x) - 1)$

- ({sec}^{2} (x) - 1)

$- (\sec^{2} (x) - 1)$

- ({sec}^{2} (x) - 1)

$- (\sec^{2} (x) - 1)$

Step 2

Apply pythagorean identity.

- {tan}^{2} (x)

$- \tan^{2} (x)$

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$