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Popular Trigonometry >

2tan^2(x)=sec^2(x)

Solution

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

Solution

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Degrees

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

sec^{2} (x)

from both sides

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

(2 tan (x) + sec (x)) (2 tan (x) - sec (x)) = 0

Solving each part separately

2 tan (x) + sec (x) = 0 or 2 tan (x) - sec (x) = 0

2 tan (x) + sec (x) = 0 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 tan (x) + sec (x) = 0

Express with sin, cos

sec (x) + 2 tan (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( x )} + 2 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )}

Simplify

\frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )} : \frac{1 + 2 sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )}

Multiply

2 \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

2 \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) 2}{cos ( x )}

= \frac{1}{cos ( x )} + \frac{2 sin ( x )}{cos ( x )}

Apply rule

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

= \frac{1 + 2 sin ( x )}{cos ( x )}

= \frac{1 + 2 sin ( x )}{cos ( x )}

\frac{1 + sin ( x ) 2}{cos ( x )} = 0

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

1 + sin (x) 2 = 0

Move

1

to the right side

1 + sin (x) 2 = 0

Subtract

1

from both sides

1 + sin (x) 2 - 1 = 0 - 1

Simplify

sin (x) 2 = - 1

sin (x) 2 = - 1

Divide both sides by

2

sin (x) 2 = - 1

Divide both sides by

2

\frac{sin ( x ) 2}{2} = \frac{- 1}{2}

Simplify

\frac{sin ( x ) 2}{2} = \frac{- 1}{2}

Simplify

\frac{sin ( x ) 2}{2} : sin (x)

\frac{sin ( x ) 2}{2}

Cancel the common factor:

2

= sin (x)

Simplify

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

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

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

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (x) = - \frac{2}{2}

sin (x) = - \frac{2}{2}

sin (x) = - \frac{2}{2}

General solutions for

sin (x) = - \frac{2}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 tan (x) - sec (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

2 tan (x) - sec (x) = 0

Express with sin, cos

- sec (x) + 2 tan (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - \frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )}

Simplify

- \frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )} : \frac{- 1 + 2 sin ( x )}{cos ( x )}

- \frac{1}{cos ( x )} + 2 \frac{sin ( x )}{cos ( x )}

Multiply

2 \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

2 \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) 2}{cos ( x )}

= - \frac{1}{cos ( x )} + \frac{2 sin ( x )}{cos ( x )}

Apply rule

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

= \frac{- 1 + 2 sin ( x )}{cos ( x )}

= \frac{- 1 + 2 sin ( x )}{cos ( x )}

\frac{- 1 + sin ( x ) 2}{cos ( x )} = 0

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

- 1 + sin (x) 2 = 0

Move

1

to the right side

- 1 + sin (x) 2 = 0

Add

1

to both sides

- 1 + sin (x) 2 + 1 = 0 + 1

Simplify

sin (x) 2 = 1

sin (x) 2 = 1

Divide both sides by

2

sin (x) 2 = 1

Divide both sides by

2

\frac{sin ( x ) 2}{2} = \frac{1}{2}

Simplify

\frac{sin ( x ) 2}{2} = \frac{1}{2}

Simplify

\frac{sin ( x ) 2}{2} : sin (x)

\frac{sin ( x ) 2}{2}

Cancel the common factor:

2

= sin (x)

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2}

General solutions for

sin (x) = \frac{2}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Combine all the solutions

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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