Trigonometry Examples

$\cos (x + \frac{π}{2}) = - \sin (x)$

Step 1

Start on the left side.

$\cos (x + \frac{π}{2})$

Step 2

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

$\cos (x) \cos (\frac{π}{2}) - \sin (x) \sin (\frac{π}{2})$

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 $0$ .

$\cos (x) \cdot 0 - \sin (x) \sin (\frac{π}{2})$

Step 3.1.2

Multiply $\cos (x)$ by $0$ .

$0 - \sin (x) \sin (\frac{π}{2})$

Step 3.1.3

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

$0 - \sin (x) \cdot 1$

Step 3.1.4

Multiply $- 1$ by $1$ .

$0 - \sin (x)$

Step 3.2

Subtract $\sin (x)$ from $0$ .

$- \sin (x)$

Step 4

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

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