What is the general solution for csc^2(x)=(1/(cos(x)))^2 ?

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

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Popular Trigonometry >

csc^2(x)=(1/(cos(x)))^2

Solution

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

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

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

Subtract

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

from both sides

csc^{2} (x) - \frac{1}{cos ^{2} ( x )} = 0

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

\frac{csc ^{2} ( x ) cos ^{2} ( x ) - 1}{cos ^{2} ( x )} = 0

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

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

Factor

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

csc^{2} (x) cos^{2} (x) - 1

Apply exponent rule:

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

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

= (cos (x) csc (x))^{2} - 1

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

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

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

Rewrite using trig identities

cos (x) csc (x) + 1

csc (x) cos (x) = cot (x)

csc (x) cos (x)

Express with sin, cos

csc (x) cos (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

= 1 + cot (x)

1 + cot (x) = 0

Move

1

to the right side

1 + cot (x) = 0

Subtract

1

from both sides

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

Simplify

cot (x) = - 1

cot (x) = - 1

General solutions for

cot (x) = - 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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

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

Rewrite using trig identities

cos (x) csc (x) - 1

csc (x) cos (x) = cot (x)

csc (x) cos (x)

Express with sin, cos

csc (x) cos (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

= - 1 + cot (x)

- 1 + cot (x) = 0

Move

1

to the right side

- 1 + cot (x) = 0

Add

1

to both sides

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

Simplify

cot (x) = 1

cot (x) = 1

General solutions for

cot (x) = 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 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

Graph

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Frequently Asked Questions (FAQ)

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