Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

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

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

$1 - \cos (t) \cos (t)$

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Add $1$ and $1$ .

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

Step 2.7

Apply pythagorean identity.

$\sin^{2} (t)$

Step 3

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

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