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Popular >

1-tan^2(x)=a^2+b^2

Solution

1 - tan^{2} (x) = a^{2} + b^{2}

Solution

x = arctan (- a^{2} - b^{2} + 1) + πn, x = arctan (- - a^{2} - b^{2} + 1) + πn

Solution steps

1 - tan^{2} (x) = a^{2} + b^{2}

Solve by substitution

1 - tan^{2} (x) = a^{2} + b^{2}

Let:

tan (x) = u

1 - u^{2} = a^{2} + b^{2}

1 - u^{2} = a^{2} + b^{2} : u = - a^{2} - b^{2} + 1, u = - - a^{2} - b^{2} + 1

1 - u^{2} = a^{2} + b^{2}

Move

1

to the right side

1 - u^{2} = a^{2} + b^{2}

Subtract

1

from both sides

1 - u^{2} - 1 = a^{2} + b^{2} - 1

Simplify

- u^{2} = a^{2} + b^{2} - 1

- u^{2} = a^{2} + b^{2} - 1

Divide both sides by

- 1

- u^{2} = a^{2} + b^{2} - 1

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{a ^{2}}{- 1} + \frac{b ^{2}}{- 1} - \frac{1}{- 1}

Simplify

\frac{- u ^{2}}{- 1} = \frac{a ^{2}}{- 1} + \frac{b ^{2}}{- 1} - \frac{1}{- 1}

Simplify

\frac{- u ^{2}}{- 1} : u^{2}

\frac{- u ^{2}}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{u ^{2}}{1}

Apply rule

\frac{a}{1} = a

= u^{2}

Simplify

\frac{a ^{2}}{- 1} + \frac{b ^{2}}{- 1} - \frac{1}{- 1} : - a^{2} - b^{2} + 1

\frac{a ^{2}}{- 1} + \frac{b ^{2}}{- 1} - \frac{1}{- 1}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{a ^{2} + b ^{2} - 1}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{a ^{2} + b ^{2} - 1}{1}

Apply rule

\frac{a}{1} = a

= - (a^{2} + b^{2} - 1)

Distribute parentheses

= - (a^{2}) - (b^{2}) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - a^{2} - b^{2} + 1

u^{2} = - a^{2} - b^{2} + 1

u^{2} = - a^{2} - b^{2} + 1

u^{2} = - a^{2} - b^{2} + 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - a^{2} - b^{2} + 1, u = - - a^{2} - b^{2} + 1

Substitute back

u = tan (x)

tan (x) = - a^{2} - b^{2} + 1, tan (x) = - - a^{2} - b^{2} + 1

tan (x) = - a^{2} - b^{2} + 1, tan (x) = - - a^{2} - b^{2} + 1

tan (x) = - a^{2} - b^{2} + 1 : x = arctan (- a^{2} - b^{2} + 1) + πn

tan (x) = - a^{2} - b^{2} + 1

Apply trig inverse properties

tan (x) = - a^{2} - b^{2} + 1

General solutions for

tan (x) = - a^{2} - b^{2} + 1

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (- a^{2} - b^{2} + 1) + πn

x = arctan (- a^{2} - b^{2} + 1) + πn

tan (x) = - - a^{2} - b^{2} + 1 : x = arctan (- - a^{2} - b^{2} + 1) + πn

tan (x) = - - a^{2} - b^{2} + 1

Apply trig inverse properties

tan (x) = - - a^{2} - b^{2} + 1

General solutions for

tan (x) = - - a^{2} - b^{2} + 1

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (- - a^{2} - b^{2} + 1) + πn

x = arctan (- - a^{2} - b^{2} + 1) + πn

Combine all the solutions

x = arctan (- a^{2} - b^{2} + 1) + πn, x = arctan (- - a^{2} - b^{2} + 1) + πn

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