Precalculus Examples

$\cos (θ - \frac{π}{2}) = \sin (θ)$

Step 1

Start on the left side.

$\cos (θ - \frac{π}{2})$

Step 2

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$\cos (θ) \cos (\frac{π}{2}) + \sin (θ) \sin (\frac{π}{2})$

Step 3

Simplify the expression.

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Step 3.1

Simplify each term.

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Step 3.1.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\cos (θ) \cdot 0 + \sin (θ) \sin (\frac{π}{2})$

Step 3.1.2

Multiply $\cos (θ)$ by $0$ .

$0 + \sin (θ) \sin (\frac{π}{2})$

Step 3.1.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$0 + \sin (θ) \cdot 1$

Step 3.1.4

Multiply $\sin (θ)$ by $1$ .

$0 + \sin (θ)$

Step 3.2

Add $0$ and $\sin (θ)$ .

$\sin (θ)$

Step 4

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

$\cos (θ - \frac{π}{2}) = \sin (θ)$ is an identity