Trigonometry Examples

t = a \tan (θ)

$t = a \tan (θ)$ ,

\sqrt{\frac{1}{t^{2} + a^{2}}}

$\sqrt{\frac{1}{t^{2} + a^{2}}}$

Step 1

Replace the variable

t

$t$ with

a \tan (θ)

$a \tan (θ)$ in the expression.

\sqrt{\frac{1}{{(a \tan (θ))}^{2} + a^{2}}}

$\sqrt{\frac{1}{{(a \tan (θ))}^{2} + a^{2}}}$

Step 2

Simplify with factoring out.

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Step 2.1

Apply the product rule to

a \tan (θ)

$a \tan (θ)$ .

\sqrt{\frac{1}{a^{2} \tan^{2} (θ) + a^{2}}}

$\sqrt{\frac{1}{a^{2} \tan^{2} (θ) + a^{2}}}$

Step 2.2

Factor

a^{2}

$a^{2}$ out of

a^{2} \tan^{2} (θ) + a^{2}

$a^{2} \tan^{2} (θ) + a^{2}$ .

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Step 2.2.1

Factor

a^{2}

$a^{2}$ out of

a^{2} \tan^{2} (θ)

$a^{2} \tan^{2} (θ)$ .

\sqrt{\frac{1}{a^{2} (\tan^{2} (θ)) + a^{2}}}

$\sqrt{\frac{1}{a^{2} (\tan^{2} (θ)) + a^{2}}}$

Step 2.2.2

Multiply by

1

$1$ .

\sqrt{\frac{1}{a^{2} (\tan^{2} (θ)) + a^{2} \cdot 1}}

$\sqrt{\frac{1}{a^{2} (\tan^{2} (θ)) + a^{2} \cdot 1}}$

Step 2.2.3

Factor

a^{2}

$a^{2}$ out of

a^{2} (\tan^{2} (θ)) + a^{2} \cdot 1

$a^{2} (\tan^{2} (θ)) + a^{2} \cdot 1$ .

\sqrt{\frac{1}{a^{2} (\tan^{2} (θ) + 1)}}

$\sqrt{\frac{1}{a^{2} (\tan^{2} (θ) + 1)}}$

\sqrt{\frac{1}{a^{2} (\tan^{2} (θ) + 1)}}

$\sqrt{\frac{1}{a^{2} (\tan^{2} (θ) + 1)}}$

\sqrt{\frac{1}{a^{2} (\tan^{2} (θ) + 1)}}

$\sqrt{\frac{1}{a^{2} (\tan^{2} (θ) + 1)}}$

Step 3

Apply pythagorean identity.

\sqrt{\frac{1}{a^{2} \sec^{2} (θ)}}

$\sqrt{\frac{1}{a^{2} \sec^{2} (θ)}}$

Step 4

Simplify with factoring out.

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Step 4.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

\sqrt{\frac{1^{2}}{a^{2} \sec^{2} (θ)}}

$\sqrt{\frac{1^{2}}{a^{2} \sec^{2} (θ)}}$

Step 4.2

Rewrite

a^{2} \sec^{2} (θ)

$a^{2} \sec^{2} (θ)$ as

{(a \sec (θ))}^{2}

${(a \sec (θ))}^{2}$ .

\sqrt{\frac{1^{2}}{{(a \sec (θ))}^{2}}}

$\sqrt{\frac{1^{2}}{{(a \sec (θ))}^{2}}}$

\sqrt{\frac{1^{2}}{{(a \sec (θ))}^{2}}}

$\sqrt{\frac{1^{2}}{{(a \sec (θ))}^{2}}}$

Step 5

Rewrite

\frac{1^{2}}{{(a \sec (θ))}^{2}}

$\frac{1^{2}}{{(a \sec (θ))}^{2}}$ as

{(\frac{1}{a \sec (θ)})}^{2}

${(\frac{1}{a \sec (θ)})}^{2}$ .

\sqrt{{(\frac{1}{a \sec (θ)})}^{2}}

$\sqrt{{(\frac{1}{a \sec (θ)})}^{2}}$

Step 6

Pull terms out from under the radical, assuming positive real numbers.

\frac{1}{a \sec (θ)}

$\frac{1}{a \sec (θ)}$

Step 7

Separate fractions.

\frac{1}{a} \cdot \frac{1}{\sec (θ)}

$\frac{1}{a} \cdot \frac{1}{\sec (θ)}$

Step 8

Rewrite

\sec (θ)

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

\frac{1}{a} \cdot \frac{1}{\frac{1}{\cos (θ)}}

$\frac{1}{a} \cdot \frac{1}{\frac{1}{\cos (θ)}}$

Step 9

Multiply by the reciprocal of the fraction to divide by

\frac{1}{\cos (θ)}

$\frac{1}{\cos (θ)}$ .

\frac{1}{a} (1 \cos (θ))

$\frac{1}{a} (1 \cos (θ))$

Step 10

Multiply

\cos (θ)

$\cos (θ)$ by

1

$1$ .

\frac{1}{a} \cos (θ)

$\frac{1}{a} \cos (θ)$

Step 11

Combine

\frac{1}{a}

$\frac{1}{a}$ and

\cos (θ)

$\cos (θ)$ .

\frac{\cos (θ)}{a}

$\frac{\cos (θ)}{a}$