Trigonometry Examples

$\frac{1}{\cos (t)} - \cos (t) \tan^{2} (t) = \cos (t)$

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

Apply the reciprocal identity to $\sec (t)$ .

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

Step 3.2

Apply the product rule to $\frac{1}{\cos (t)}$ .

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Cancel the common factor of $\cos (t)$ .

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Step 4.1.3.1

Factor $\cos (t)$ out of $- \cos (t)$ .

$\frac{1}{\cos (t)} + \cos (t) \cdot - 1 \frac{1}{{\cos (t)}^{2}} - \cos (t) \cdot - 1$

Step 4.1.3.2

Factor $\cos (t)$ out of ${\cos (t)}^{2}$ .

$\frac{1}{\cos (t)} + \cos (t) \cdot - 1 \frac{1}{\cos (t) \cos (t)} - \cos (t) \cdot - 1$

Step 4.1.3.3

Cancel the common factor.

$\frac{1}{\cos (t)} + \cos (t) \cdot - 1 \frac{1}{\cos (t) \cos (t)} - \cos (t) \cdot - 1$

Step 4.1.3.4

Rewrite the expression.

$\frac{1}{\cos (t)} - 1 \frac{1}{\cos (t)} - \cos (t) \cdot - 1$

Step 4.1.4

Multiply $- \cos (t) \cdot - 1$ .

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Step 4.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{\cos (t)} - 1 \frac{1}{\cos (t)} + 1 \cos (t)$

Step 4.1.4.2

Multiply $\cos (t)$ by $1$ .

$\frac{1}{\cos (t)} - 1 \frac{1}{\cos (t)} + \cos (t)$

Step 4.1.5

Rewrite $- 1 \frac{1}{\cos (t)}$ as $- \frac{1}{\cos (t)}$ .

$\frac{1}{\cos (t)} - \frac{1}{\cos (t)} + \cos (t)$

Step 4.2

Combine the numerators over the common denominator.

$\cos (t) + \frac{1 - 1}{\cos (t)}$

Step 4.3

Subtract $1$ from $1$ .

$\cos (t) + \frac{0}{\cos (t)}$

Step 4.4

Divide $0$ by $\cos (t)$ .

$\cos (t) + 0$

Step 4.5

Add $\cos (t)$ and $0$ .

$\cos (t)$

Step 5

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

$\frac{1}{\cos (t)} - \cos (t) \tan^{2} (t) = \cos (t)$ is an identity