Trigonometry Examples

$\frac{1}{125 \tan^{3} (θ)} = \frac{(\cot^{3} (θ))}{125}$

Step 1

Start on the left side.

$\frac{1}{125 \tan^{3} (θ)}$

Step 2

Convert to sines and cosines.

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Step 2.1

Write $\tan (θ)$ in sines and cosines using the quotient identity.

$\frac{1}{125 {(\frac{\sin (θ)}{\cos (θ)})}^{3}}$

Step 2.2

Apply the product rule to $\frac{\sin (θ)}{\cos (θ)}$ .

$\frac{1}{125 \frac{\sin^{3} (θ)}{\cos^{3} (θ)}}$

Step 3

Simplify.

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Step 3.1

Combine $125$ and $\frac{{\sin (θ)}^{3}}{{\cos (θ)}^{3}}$ .

$\frac{1}{\frac{125 {\sin (θ)}^{3}}{{\cos (θ)}^{3}}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{{\cos (θ)}^{3}}{125 {\sin (θ)}^{3}}$

Step 3.3

Multiply $\frac{{\cos (θ)}^{3}}{125 {\sin (θ)}^{3}}$ by $1$ .

$\frac{\cos^{3} (θ)}{125 \sin^{3} (θ)}$

Step 4

Rewrite $\frac{\cos^{3} (θ)}{125 \sin^{3} (θ)}$ as $\frac{(\cot^{3} (θ))}{125}$ .

$\frac{(\cot^{3} (θ))}{125}$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{1}{125 \tan^{3} (θ)} = \frac{\cot^{3} (θ)}{125}$ is an identity