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

The general solution for 2sec(x)tan(x)-sec^2(x)=0 is x= pi/6+2pin,x=(5pi)/6+2pin

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

Solution

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

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

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

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

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

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

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

Factor out common term

sec (x)

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

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

Solving each part separately

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

sec (x) = 0 :

No Solution

sec (x) = 0

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

No Solution

2 tan (x) - sec (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 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 \cdot \frac{sin ( x )}{cos ( x )}

Simplify

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

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

Multiply

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

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

Multiply fractions:

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

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

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

- 1 + 2 sin (x) = 0

Move

1

to the right side

- 1 + 2 sin (x) = 0

Add

1

to both sides

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

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

sin (x) = \frac{1}{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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Popular Examples

csc^2(x)=cot(x)+1

csc^{2} (x) = cot (x) + 1

csc^2(x)=2cot(x)

csc^{2} (x) = 2 cot (x)

21csc^2(2x)-5=23

21 csc^{2} (2 x) - 5 = 23

4sin^2(x/3)=1

4 sin^{2} (\frac{x}{3}) = 1

cos(2x-pi/4)=(sqrt(2))/2

cos (2 x - \frac{π}{4}) = \frac{2}{2}

Frequently Asked Questions (FAQ)

What is the general solution for 2sec(x)tan(x)-sec^2(x)=0 ?
The general solution for 2sec(x)tan(x)-sec^2(x)=0 is x= pi/6+2pin,x=(5pi)/6+2pin