What is the general solution for sin^2(x)+cos^2(x)=sin(x)csc^2(x) ?

The general solution for sin^2(x)+cos^2(x)=sin(x)csc^2(x) is x= pi/2+2pin

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sin^2(x)+cos^2(x)=sin(x)csc^2(x)

Solution

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

Solution

x = \frac{π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

sin (x) csc^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Multiply:

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

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

Cancel the common factor:

sin (x)

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

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

Use the Pythagorean identity:

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

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

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

Multiply both sides by

sin (x)

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

Multiply both sides by

sin (x)

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

Simplify

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= - 1

Simplify

1 \cdot sin (x) : sin (x)

1 \cdot sin (x)

Multiply:

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

= sin (x)

Simplify

0 \cdot sin (x) : 0

0 \cdot sin (x)

Apply rule

0 \cdot a = 0

= 0

- 1 + sin (x) = 0

- 1 + sin (x) = 0

- 1 + sin (x) = 0

Move

1

to the right side

- 1 + sin (x) = 0

Add

1

to both sides

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

Simplify

sin (x) = 1

sin (x) = 1

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

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

and compare to zero

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = 1

General solutions for

sin (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

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