Trigonometry Examples

$\sin^{2} (A) - \cos^{2} (A) = 2 \cos^{2} (A) \tan^{2} (A) - 1$

Step 1

Replace the $\sin^{2} (A)$ with $1 - \cos^{2} (A)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$(1 - \cos^{2} (A)) - \cos^{2} (A) = 2 \cos^{2} (A) \tan^{2} (A) - 1$

Step 2

Subtract $\cos^{2} (A)$ from $- \cos^{2} (A)$ .

$1 - 2 \cos^{2} (A) = 2 \cos^{2} (A) \tan^{2} (A) - 1$

Step 3

Reorder the polynomial.

$- 2 \cos^{2} (A) + 1 = 2 \cos^{2} (A) \tan^{2} (A) - 1$

Step 4

Simplify the right side.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite $\tan (A)$ in terms of sines and cosines.

$- 2 \cos^{2} (A) + 1 = 2 \cos^{2} (A) {(\frac{\sin (A)}{\cos (A)})}^{2} - 1$

Step 4.1.2

Apply the product rule to $\frac{\sin (A)}{\cos (A)}$ .

$- 2 \cos^{2} (A) + 1 = 2 \cos^{2} (A) \frac{\sin^{2} (A)}{\cos^{2} (A)} - 1$

Step 4.1.3

Cancel the common factor of $\cos^{2} (A)$ .

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Step 4.1.3.1

Factor $\cos^{2} (A)$ out of $2 \cos^{2} (A)$ .

$- 2 \cos^{2} (A) + 1 = \cos^{2} (A) \cdot 2 \frac{\sin^{2} (A)}{\cos^{2} (A)} - 1$

Step 4.1.3.2

Cancel the common factor.

$- 2 \cos^{2} (A) + 1 = \cos^{2} (A) \cdot 2 \frac{\sin^{2} (A)}{\cos^{2} (A)} - 1$

Step 4.1.3.3

Rewrite the expression.

$- 2 \cos^{2} (A) + 1 = 2 \sin^{2} (A) - 1$

Step 5

Move all the expressions to the left side of the equation.

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Step 5.1

Subtract $2 \sin^{2} (A)$ from both sides of the equation.

$- 2 \cos^{2} (A) + 1 - 2 \sin^{2} (A) = - 1$

Step 5.2

Add $1$ to both sides of the equation.

$- 2 \cos^{2} (A) + 1 - 2 \sin^{2} (A) + 1 = 0$

Step 6

Simplify $- 2 \cos^{2} (A) + 1 - 2 \sin^{2} (A) + 1$ .

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Step 6.1

Move $- 2 \sin^{2} (A)$ .

$- 2 \cos^{2} (A) - 2 \sin^{2} (A) + 1 + 1 = 0$

Step 6.2

Factor $- 2$ out of $- 2 \cos^{2} (A)$ .

$- 2 (\cos^{2} (A)) - 2 \sin^{2} (A) + 1 + 1 = 0$

Step 6.3

Factor $- 2$ out of $- 2 \sin^{2} (A)$ .

$- 2 (\cos^{2} (A)) - 2 (\sin^{2} (A)) + 1 + 1 = 0$

Step 6.4

Factor $- 2$ out of $- 2 (\cos^{2} (A)) - 2 (\sin^{2} (A))$ .

$- 2 (\cos^{2} (A) + \sin^{2} (A)) + 1 + 1 = 0$

Step 6.5

Rearrange terms.

$- 2 (\sin^{2} (A) + \cos^{2} (A)) + 1 + 1 = 0$

Step 6.6

Apply pythagorean identity.

$- 2 \cdot 1 + 1 + 1 = 0$

Step 6.7

Multiply $- 2$ by $1$ .

$- 2 + 1 + 1 = 0$

Step 6.8

Add $- 2$ and $1$ .

$- 1 + 1 = 0$

Step 6.9

Add $- 1$ and $1$ .

$0 = 0$

Step 7

Since $0 = 0$ , the equation will always be true for any value of $A$ .

All real numbers

Step 8

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$