Trigonometry Examples

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

Step 1

Replace the variable $t$ with $a \tan (θ)$ in the expression.

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

Step 2

Simplify with factoring out.

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Step 2.1

Apply the product rule to $a \tan (θ)$ .

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

Step 2.2

Factor $a^{2}$ out of $a^{2} \tan^{2} (θ) + a^{2}$ .

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Step 2.2.1

Factor $a^{2}$ out of $a^{2} \tan^{2} (θ)$ .

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

Step 2.2.2

Multiply by $1$ .

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

Step 2.2.3

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

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

Step 3

Apply pythagorean identity.

$\sqrt{\frac{1}{a^{2} \sec^{2} (θ)}}$

Step 4

Simplify with factoring out.

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Step 4.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.2

Rewrite $a^{2} \sec^{2} (θ)$ as ${(a \sec (θ))}^{2}$ .

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

Step 5

Rewrite $\frac{1^{2}}{{(a \sec (θ))}^{2}}$ as ${(\frac{1}{a \sec (θ)})}^{2}$ .

$\sqrt{{(\frac{1}{a \sec (θ)})}^{2}}$

Step 6

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

$\frac{1}{a \sec (θ)}$

Step 7

Separate fractions.

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

Step 8

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

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

Step 9

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

$\frac{1}{a} (1 \cos (θ))$

Step 10

Multiply $\cos (θ)$ by $1$ .

$\frac{1}{a} \cos (θ)$

Step 11

Combine $\frac{1}{a}$ and $\cos (θ)$ .

$\frac{\cos (θ)}{a}$