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

The general solution for cos^2(x)tan^2(x)csc^2(x)=sin(x) is No Solution for x\in\mathbb{R}

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

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

sin (x)

from both sides

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin^{2} (x)

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

Cancel the common factor:

cos^{2} (x)

= 1

= - sin (x) + 1

= - sin (x) + 1

1 - sin (x) = 0

Move

1

to the right side

1 - sin (x) = 0

Subtract

1

from both sides

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

Simplify

- sin (x) = - 1

- sin (x) = - 1

Divide both sides by

- 1

- sin (x) = - 1

Divide both sides by

- 1

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

Simplify

sin (x) = 1

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

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution for x \in R

Graph

Popular Examples

cos(a)-sin(a)=cot(a)cos(a)

cos (a) - sin (a) = cot (a) cos (a)

15cos((2pi)/(365)t)+43=30

15 cos (\frac{2 π}{365} t) + 43 = 30

tan^2(x)=2sec(x)-1

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

cos(z)=1

cos (z) = 1

cos(z)=5

cos (z) = 5

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)tan^2(x)csc^2(x)=sin(x) ?
The general solution for cos^2(x)tan^2(x)csc^2(x)=sin(x) is No Solution for x\in\mathbb{R}