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

The general solution for sec^2(x)= 1/(sin^2(x)) is x=(3pi)/4+pin,x= pi/4+pin

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sec^2(x)= 1/(sin^2(x))

Solution

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

Solution

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

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Factor

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

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

Apply exponent rule:

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

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

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

Apply Difference of Two Squares Formula:

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

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

= (sec (x) sin (x) + 1) (sec (x) sin (x) - 1)

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

Solving each part separately

sec (x) sin (x) + 1 = 0 or sec (x) sin (x) - 1 = 0

sec (x) sin (x) + 1 = 0 : x = \frac{3 π}{4} + πn

sec (x) sin (x) + 1 = 0

Rewrite using trig identities

sec (x) sin (x) + 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Express with sin, cos

sec (x) sin (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

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

Use the basic trigonometric identity:

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

= tan (x)

= 1 + tan (x)

1 + tan (x) = 0

Move

1

to the right side

1 + tan (x) = 0

Subtract

1

from both sides

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

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

sec (x) sin (x) - 1 = 0 : x = \frac{π}{4} + πn

sec (x) sin (x) - 1 = 0

Rewrite using trig identities

sec (x) sin (x) - 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Express with sin, cos

sec (x) sin (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

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

Use the basic trigonometric identity:

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

= tan (x)

= - 1 + tan (x)

- 1 + tan (x) = 0

Move

1

to the right side

- 1 + tan (x) = 0

Add

1

to both sides

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

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

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

Popular Examples

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2 sin^{2} (x) + sin (x) - 1 = 0, 0 \leq x < 2 π

arctan(x)=(7.8)/(-4.42)

arctan (x) = \frac{7.8}{- 4.42}

-sin(x+5)=0

- sin (x + 5^{\circ}) = 0

cos(2pix)*2pi+2pi=0

cos (2 π x) \cdot 2 π + 2 π = 0

solvefor x,sin(y)=ycos(x)solve for

x, sin (y) = y cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)= 1/(sin^2(x)) ?
The general solution for sec^2(x)= 1/(sin^2(x)) is x=(3pi)/4+pin,x= pi/4+pin