Trigonometry Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $\tan (t)$ in terms of sines and cosines.

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

Step 1.1.2

Multiply $- \sin (t) \frac{\sin (t)}{\cos (t)}$ .

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Step 1.1.2.1

Combine $\frac{\sin (t)}{\cos (t)}$ and $\sin (t)$ .

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

Step 1.1.2.2

Raise $\sin (t)$ to the power of $1$ .

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

Step 1.1.2.3

Raise $\sin (t)$ to the power of $1$ .

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

Step 1.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.2.5

Add $1$ and $1$ .

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Apply pythagorean identity.

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

Step 3

Cancel the common factor of $\cos^{2} (t)$ and $\cos (t)$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.2.4

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

$\cos (t)$