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

The general solution for sin^2(x)sec(x)=tan^2(x) is x=2pin,x=pi+2pin

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Popular Trigonometry >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

tan^{2} (x)

from both sides

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

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

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

Multiply fractions:

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

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

Multiply:

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

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

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

Least Common Multiplier of

cos^{2} (x), cos (x) : cos^{2} (x)

cos^{2} (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos^{2} (x)

cos (x)

= cos^{2} (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

cos^{2} (x)

For

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

multiply the denominator and numerator by

cos (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Factor

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

- sin^{2} (x) + cos (x) sin^{2} (x)

Factor out common term

sin^{2} (x)

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

sin^{2} (x) (cos (x) - 1) = 0

Solving each part separately

sin^{2} (x) = 0 or cos (x) - 1 = 0

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Move

1

to the right side

cos (x) - 1 = 0

Add

1

to both sides

cos (x) - 1 + 1 = 0 + 1

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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

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, x = π + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)sec(x)=tan^2(x) ?
The general solution for sin^2(x)sec(x)=tan^2(x) is x=2pin,x=pi+2pin