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

The general solution for 1/2 tan^2(x)=1-sec(x) is x=2pin,x=1.91063…+2pin,x=-1.91063…+2pin

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

Solution

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

Solution

x = 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1 - sec (x)

from both sides

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

Simplify

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

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

\frac{1}{2} tan^{2} (x) = \frac{tan ^{2} ( x )}{2}

\frac{1}{2} tan^{2} (x)

Multiply fractions:

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

= \frac{1 \cdot tan ^{2} ( x )}{2}

Multiply:

1 \cdot tan^{2} (x) = tan^{2} (x)

= \frac{tan ^{2} ( x )}{2}

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

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}, sec (x) = \frac{sec ( x ) 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

- 2 + sec^{2} (x) - 1 + 2 sec (x) : sec^{2} (x) + 2 sec (x) - 3

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

Group like terms

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

Subtract the numbers:

- 2 - 1 = - 3

= sec^{2} (x) + 2 sec (x) - 3

= sec^{2} (x) + 2 sec (x) - 3

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

Solve by substitution

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

Let:

sec (x) = u

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

- 3 + u^{2} + 2 u = 0 : u = 1, u = - 3

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

Write in the standard form

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

u^{2} + 2 u - 3 = 0

Solve with the quadratic formula

u^{2} + 2 u - 3 = 0

Quadratic Equation Formula:

For

a = 1, b = 2, c = - 3

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 3 )}{2 \cdot 1}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 3 )}{2 \cdot 1}

2^{2} - 4 \cdot 1 \cdot (- 3) = 4

2^{2} - 4 \cdot 1 \cdot (- 3)

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 1 \cdot 3

Multiply the numbers:

4 \cdot 1 \cdot 3 = 12

= 2^{2} + 12

2^{2} = 4

= 4 + 12

Add the numbers:

4 + 12 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

u_{1, 2} = \frac{- 2 \pm 4}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 2 + 4}{2 \cdot 1}, u_{2} = \frac{- 2 - 4}{2 \cdot 1}

u = \frac{- 2 + 4}{2 \cdot 1} : 1

\frac{- 2 + 4}{2 \cdot 1}

Add/Subtract the numbers:

- 2 + 4 = 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{- 2 - 4}{2 \cdot 1} : - 3

\frac{- 2 - 4}{2 \cdot 1}

Subtract the numbers:

- 2 - 4 = - 6

= \frac{- 6}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6}{2}

Apply the fraction rule:

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

= - \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3

The solutions to the quadratic equation are:

u = 1, u = - 3

Substitute back

u = sec (x)

sec (x) = 1, sec (x) = - 3

sec (x) = 1, sec (x) = - 3

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

sec (x) = - 3 : x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

sec (x) = - 3

Apply trig inverse properties

sec (x) = - 3

General solutions for

sec (x) = - 3

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

Combine all the solutions

x = 2 πn, x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

Show solutions in decimal form

x = 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

Popular Examples

sin(2x)=sin^2(x)

sin (2 x) = sin^{2} (x)

2sec^2(x)+3(1/(cos(x)))=2

2 sec^{2} (x) + 3 (\frac{1}{cos ( x )}) = 2

cot^2(θ)-3=0

cot^{2} (θ) - 3 = 0

tan(θ)=2,sin(θ)

tan (θ) = 2, sin (θ)

7sin^2(x)+14sin(x)+7=0

7 sin^{2} (x) + 14 sin (x) + 7 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 1/2 tan^2(x)=1-sec(x) ?
The general solution for 1/2 tan^2(x)=1-sec(x) is x=2pin,x=1.91063…+2pin,x=-1.91063…+2pin