Trigonometry Examples

$\cos^{2} (x) - \sin^{2} (x)$

Step 1

Interchange the variables.

$x = \cos^{2} (y) - \sin^{2} (y)$

Step 2

Solve for $y$ .

Tap for more steps...

Rewrite the equation as $\cos^{2} (y) - \sin^{2} (y) = x$ .

$\cos^{2} (y) - \sin^{2} (y) = x$

Subtract $x$ from both sides of the equation.

$\cos^{2} (y) - \sin^{2} (y) - x = 0$

Apply the cosine double-angle identity.

$\cos (2 y) - x = 0$

Solve for $y$ .

Tap for more steps...

Add $x$ to both sides of the equation.

$\cos (2 y) = x$

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$2 y = \arccos (x)$

Divide each term in $2 y = \arccos (x)$ by $2$ and simplify.

Tap for more steps...

Divide each term in $2 y = \arccos (x)$ by $2$ .

$\frac{2 y}{2} = \frac{\arccos (x)}{2}$

Simplify the left side.

Tap for more steps...

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$\frac{2 y}{2} = \frac{\arccos (x)}{2}$

Divide $y$ by $1$ .

$y = \frac{\arccos (x)}{2}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{\arccos (x)}{2}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arccos (x)}{2}$ is the inverse of $f (x) = \cos^{2} (x) - \sin^{2} (x)$ .

Tap for more steps...

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Tap for more steps...

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (\cos^{2} (x) - \sin^{2} (x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\cos^{2} (x) - \sin^{2} (x)) = \frac{\arccos (\cos^{2} (x) - \sin^{2} (x))}{2}$

Apply the cosine double-angle identity.

$f^{- 1} (\cos^{2} (x) - \sin^{2} (x)) = \frac{\arccos (\cos (2 x))}{2}$

Evaluate $f (f^{- 1} (x))$ .

Tap for more steps...

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (\frac{\arccos (x)}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (x)}{2}) = \cos^{2} (\frac{\arccos (x)}{2}) - \sin^{2} (\frac{\arccos (x)}{2})$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (\frac{\arccos (x)}{2})$ and $b = \sin (\frac{\arccos (x)}{2})$ .

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

Expand $(\cos (\frac{\arccos (x)}{2}) + \sin (\frac{\arccos (x)}{2})) (\cos (\frac{\arccos (x)}{2}) - \sin (\frac{\arccos (x)}{2}))$ using the FOIL Method.

Tap for more steps...

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify terms.

Tap for more steps...

Combine the opposite terms in $\cos (\frac{\arccos (x)}{2}) \cos (\frac{\arccos (x)}{2}) + \cos (\frac{\arccos (x)}{2}) (- \sin (\frac{\arccos (x)}{2})) + \sin (\frac{\arccos (x)}{2}) \cos (\frac{\arccos (x)}{2}) + \sin (\frac{\arccos (x)}{2}) (- \sin (\frac{\arccos (x)}{2}))$ .

Tap for more steps...

Reorder the factors in the terms $\cos (\frac{\arccos (x)}{2}) (- \sin (\frac{\arccos (x)}{2}))$ and $\sin (\frac{\arccos (x)}{2}) \cos (\frac{\arccos (x)}{2})$ .

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

Add $- \cos (\frac{\arccos (x)}{2}) \sin (\frac{\arccos (x)}{2})$ and $\cos (\frac{\arccos (x)}{2}) \sin (\frac{\arccos (x)}{2})$ .

$f (\frac{\arccos (x)}{2}) = \cos (\frac{\arccos (x)}{2}) \cos (\frac{\arccos (x)}{2}) + 0 + \sin (\frac{\arccos (x)}{2}) (- \sin (\frac{\arccos (x)}{2}))$

Add $\cos (\frac{\arccos (x)}{2}) \cos (\frac{\arccos (x)}{2})$ and $0$ .

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

Simplify each term.

Tap for more steps...

Multiply $\cos (\frac{\arccos (x)}{2}) \cos (\frac{\arccos (x)}{2})$ .

Tap for more steps...

Raise $\cos (\frac{\arccos (x)}{2})$ to the power of $1$ .

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

Raise $\cos (\frac{\arccos (x)}{2})$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{\arccos (x)}{2}) = {\cos (\frac{\arccos (x)}{2})}^{1 + 1} + \sin (\frac{\arccos (x)}{2}) (- \sin (\frac{\arccos (x)}{2}))$

Add $1$ and $1$ .

$f (\frac{\arccos (x)}{2}) = \cos^{2} (\frac{\arccos (x)}{2}) + \sin (\frac{\arccos (x)}{2}) (- \sin (\frac{\arccos (x)}{2}))$

Rewrite using the commutative property of multiplication.

$f (\frac{\arccos (x)}{2}) = \cos^{2} (\frac{\arccos (x)}{2}) - \sin (\frac{\arccos (x)}{2}) \sin (\frac{\arccos (x)}{2})$

Multiply $- \sin (\frac{\arccos (x)}{2}) \sin (\frac{\arccos (x)}{2})$ .

Tap for more steps...

Raise $\sin (\frac{\arccos (x)}{2})$ to the power of $1$ .

$f (\frac{\arccos (x)}{2}) = \cos^{2} (\frac{\arccos (x)}{2}) - (\sin (\frac{\arccos (x)}{2}) \sin (\frac{\arccos (x)}{2}))$

Raise $\sin (\frac{\arccos (x)}{2})$ to the power of $1$ .

$f (\frac{\arccos (x)}{2}) = \cos^{2} (\frac{\arccos (x)}{2}) - (\sin (\frac{\arccos (x)}{2}) \sin (\frac{\arccos (x)}{2}))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{\arccos (x)}{2}) = \cos^{2} (\frac{\arccos (x)}{2}) - {\sin (\frac{\arccos (x)}{2})}^{1 + 1}$

Add $1$ and $1$ .

$f (\frac{\arccos (x)}{2}) = \cos^{2} (\frac{\arccos (x)}{2}) - \sin^{2} (\frac{\arccos (x)}{2})$

Apply the cosine double-angle identity.

$f (\frac{\arccos (x)}{2}) = \cos (2 (\frac{\arccos (x)}{2}))$

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$f (\frac{\arccos (x)}{2}) = \cos (2 (\frac{\arccos (x)}{2}))$

Rewrite the expression.

$f (\frac{\arccos (x)}{2}) = \cos (\arccos (x))$

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (x)}{2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (x)}{2}$ is the inverse of $f (x) = \cos^{2} (x) - \sin^{2} (x)$ .

$f^{- 1} (x) = \frac{\arccos (x)}{2}$