Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the sum of angles identity.

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

Step 3

Apply the sum of angles identity.

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

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

$0 + \cos (x) \cdot 1 + \sin (\frac{π}{2}) \cos (- x) + \cos (\frac{π}{2}) \sin (- x)$

Step 4.1.4

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

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

Step 4.1.5

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

$0 + \cos (x) + 1 \cos (- x) + \cos (\frac{π}{2}) \sin (- x)$

Step 4.1.6

Multiply $\cos (- x)$ by $1$ .

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

Step 4.1.7

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 4.1.8

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

$0 + \cos (x) + \cos (x) + 0 \sin (- x)$

Step 4.1.9

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$0 + \cos (x) + \cos (x) + 0 (- \sin (x))$

Step 4.1.10

Multiply $0 (- \sin (x))$ .

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Step 4.1.10.1

Multiply $- 1$ by $0$ .

$0 + \cos (x) + \cos (x) + 0 \sin (x)$

Step 4.1.10.2

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

$0 + \cos (x) + \cos (x) + 0$

Step 4.2

Combine the opposite terms in $0 + \cos (x) + \cos (x) + 0$ .

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Step 4.2.1

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

$\cos (x) + \cos (x) + 0$

Step 4.2.2

Add $\cos (x) + \cos (x)$ and $0$ .

$\cos (x) + \cos (x)$

Step 4.3

Add $\cos (x)$ and $\cos (x)$ .

$2 \cos (x)$

Step 5

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

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