What is the general solution for sec(x)=sqrt(1-tan^2(x)) ?

The general solution for sec(x)=sqrt(1-tan^2(x)) is x=2pin

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sec(x)=sqrt(1-tan^2(x))

Solution

sec (x) = 1 - tan^{2} (x)

Solution

x = 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n

Solution steps

sec (x) = 1 - tan^{2} (x)

Subtract

1 - tan^{2} (x)

from both sides

sec (x) - 1 - tan^{2} (x) = 0

Rewrite using trig identities

sec (x) - 1 - tan^{2} (x)

Use the Pythagorean identity:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= sec (x) - 1 - (sec^{2} (x) - 1)

Expand

1 - (sec^{2} (x) - 1) : - sec^{2} (x) + 2

1 - (sec^{2} (x) - 1)

- (sec^{2} (x) - 1) : - sec^{2} (x) + 1

- (sec^{2} (x) - 1)

Distribute parentheses

= - (sec^{2} (x)) - (- 1)

Apply minus-plus rules

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

= - sec^{2} (x) + 1

= 1 - sec^{2} (x) + 1

Simplify

1 - sec^{2} (x) + 1 : - sec^{2} (x) + 2

1 - sec^{2} (x) + 1

Group like terms

= - sec^{2} (x) + 1 + 1

Add the numbers:

1 + 1 = 2

= - sec^{2} (x) + 2

= - sec^{2} (x) + 2

= sec (x) - - sec^{2} (x) + 2

sec (x) - 2 - sec^{2} (x) = 0

Solve by substitution

sec (x) - 2 - sec^{2} (x) = 0

Let:

sec (x) = u

u - 2 - u^{2} = 0

u - 2 - u^{2} = 0 : u = 1

u - 2 - u^{2} = 0

Remove square roots

u - 2 - u^{2} = 0

Subtract

u

from both sides

u - 2 - u^{2} - u = 0 - u

Simplify

- 2 - u^{2} = - u

Square both sides:

2 - u^{2} = u^{2}

u - 2 - u^{2} = 0

(- 2 - u^{2})^{2} = (- u)^{2}

Expand

(- 2 - u^{2})^{2} : 2 - u^{2}

(- 2 - u^{2})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2 - u^{2})^{2} = (2 - u^{2})^{2}

= (2 - u^{2})^{2}

Apply radical rule:

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

= ((2 - u^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (2 - u^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2 - u^{2}

Expand

(- u)^{2} : u^{2}

(- u)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- u)^{2} = u^{2}

= u^{2}

2 - u^{2} = u^{2}

2 - u^{2} = u^{2}

2 - u^{2} = u^{2}

Solve

2 - u^{2} = u^{2} : u = 1, u = - 1

2 - u^{2} = u^{2}

Move

2

to the right side

2 - u^{2} = u^{2}

Subtract

2

from both sides

2 - u^{2} - 2 = u^{2} - 2

Simplify

- u^{2} = u^{2} - 2

- u^{2} = u^{2} - 2

Move

u^{2}

to the left side

- u^{2} = u^{2} - 2

Subtract

u^{2}

from both sides

- u^{2} - u^{2} = u^{2} - 2 - u^{2}

Simplify

- 2 u^{2} = - 2

- 2 u^{2} = - 2

Divide both sides by

- 2

- 2 u^{2} = - 2

Divide both sides by

- 2

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

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

u = 1, u = - 1

Verify Solutions:

u = 1

True

, u = - 1

False

Check the solutions by plugging them into

u - 2 - u^{2} = 0

Remove the ones that don't agree with the equation.

Plug in

u = 1 :

True

1 - 2 - 1^{2} = 0

1 - 2 - 1^{2} = 0

1 - 2 - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 2 - 1

2 - 1 = 1

2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

Apply rule

1 = 1

= 1

= 1 - 1

Subtract the numbers:

1 - 1 = 0

= 0

0 = 0

True

Plug in

u = - 1 :

False

(- 1) - 2 - (- 1)^{2} = 0

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

(- 1) - 2 - (- 1)^{2}

Remove parentheses:

(- a) = - a

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

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

2 - (- 1)^{2}

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

Apply rule

1 = 1

= 1

= - 1 - 1

Subtract the numbers:

- 1 - 1 = - 2

= - 2

- 2 = 0

False

The solution is

u = 1

Substitute back

u = sec (x)

sec (x) = 1

sec (x) = 1

sec (x) = 1 : x = 2 πn

sec (x) = 1

General solutions for

sec (x) = 1

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Combine all the solutions

x = 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sec(x)=sqrt(1-tan^2(x)) ?
The general solution for sec(x)=sqrt(1-tan^2(x)) is x=2pin