What is the general solution for tan^2(x)=sec(x)-1,0<= x<= 2pi ?

Q: What is the general solution for tan^2(x)=sec(x)-1,0<= x<= 2pi ?

The general solution for tan^2(x)=sec(x)-1,0<= x<= 2pi is x=0,x=2pi

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tan^2(x)=sec(x)-1,0<= x<= 2pi

Solution

tan^{2} (x) = sec (x) - 1, 0 \leq x \leq 2 π

Solution

x = 0, x = 2 π

Degrees

x = 0^{\circ}, x = 36 0^{\circ}

Solution steps

tan^{2} (x) = sec (x) - 1, 0 \leq x \leq 2 π

Subtract

sec (x) - 1

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Solve by substitution

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

Let:

sec (x) = u

- u + u^{2} = 0

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

- u + u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

u^{2} - u = 0

Solve with the quadratic formula

u^{2} - u = 0

Quadratic Equation Formula:

For

a = 1, b = - 1, c = 0

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 1)^{2} - 4 \cdot 1 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 1 \cdot 0

(- 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

4 \cdot 1 \cdot 0 = 0

4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- ( - 1 ) \pm 1}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 1}{2 \cdot 1}, u_{2} = \frac{- ( - 1 ) - 1}{2 \cdot 1}

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

\frac{- ( - 1 ) + 1}{2 \cdot 1}

Apply rule

- (- a) = a

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

Add the numbers:

1 + 1 = 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 1 ) - 1}{2 \cdot 1} : 0

\frac{- ( - 1 ) - 1}{2 \cdot 1}

Apply rule

- (- a) = a

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

Subtract the numbers:

1 - 1 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = 1, u = 0

Substitute back

u = sec (x)

sec (x) = 1, sec (x) = 0

sec (x) = 1, sec (x) = 0

sec (x) = 1, 0 \leq x \leq 2 π : x = 0, x = 2 π

sec (x) = 1, 0 \leq x \leq 2 π

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

Solutions for the range

0 \leq x \leq 2 π

x = 0, x = 2 π

sec (x) = 0, 0 \leq x \leq 2 π :

No Solution

sec (x) = 0, 0 \leq x \leq 2 π

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

Combine all the solutions

x = 0, x = 2 π

Popular Examples

sin^2(θ)=2sin^2(θ/2)

sin^{2} (θ) = 2 sin^{2} (\frac{θ}{2})

cos(x)=sqrt(cos(x))

cos (x) = cos (x)

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

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

2cos^3(x)+cos^2(x)-cos(x)=0

2 cos^{3} (x) + cos^{2} (x) - cos (x) = 0

0.26=sin(0.25pit)

0.26 = sin (0.25 π t)

Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)=sec(x)-1,0<= x<= 2pi ?
The general solution for tan^2(x)=sec(x)-1,0<= x<= 2pi is x=0,x=2pi