Precalculus Examples

$4 \sin^{2} (x) + 7 \cos^{2} (x) = 7 - 3 \sin^{2} (x)$

Step 1

Start on the left side.

$4 \sin^{2} (x) + 7 \cos^{2} (x)$

Step 2

Apply Pythagorean identity in reverse.

$4 (1 - \cos^{2} (x)) + 7 \cos^{2} (x)$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

$4 \cdot 1 + 4 (- {\cos (x)}^{2}) + 7 {\cos (x)}^{2}$

Step 3.1.2

Multiply $4$ by $1$ .

$4 + 4 (- {\cos (x)}^{2}) + 7 {\cos (x)}^{2}$

Step 3.1.3

Multiply $- 1$ by $4$ .

$4 - 4 {\cos (x)}^{2} + 7 {\cos (x)}^{2}$

Step 3.2

Add $- 4 {\cos (x)}^{2}$ and $7 {\cos (x)}^{2}$ .

$4 + 3 \cos^{2} (x)$

Step 4

Apply Pythagorean identity in reverse.

$4 + 3 (1 - \sin^{2} (x))$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$4 + 3 \cdot 1 + 3 (- {\sin (x)}^{2})$

Step 5.1.2

Multiply $3$ by $1$ .

$4 + 3 + 3 (- {\sin (x)}^{2})$

Step 5.1.3

Multiply $- 1$ by $3$ .

$4 + 3 - 3 {\sin (x)}^{2}$

Step 5.2

Add $4$ and $3$ .

$7 - 3 \sin^{2} (x)$

Step 6

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

$4 \sin^{2} (x) + 7 \cos^{2} (x) = 7 - 3 \sin^{2} (x)$ is an identity