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

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

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Popular >

(sin^2(x)-tan^2(x))/(csc(x))=sin(x)

Solution

\frac{sin ^{2} ( x ) - tan ^{2} ( x )}{csc ( x )} = sin (x)

Solution

No Solution for x \in R

Solution steps

\frac{sin ^{2} ( x ) - tan ^{2} ( x )}{csc ( x )} = sin (x)

Subtract

sin (x)

from both sides

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

Simplify

\frac{sin ^{2} ( x ) - tan ^{2} ( x )}{csc ( x )} - sin (x) : \frac{sin ^{2} ( x ) - tan ^{2} ( x ) - sin ( x ) csc ( x )}{csc ( x )}

\frac{sin ^{2} ( x ) - tan ^{2} ( x )}{csc ( x )} - sin (x)

Convert element to fraction:

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

= \frac{sin ^{2} ( x ) - tan ^{2} ( x )}{csc ( x )} - \frac{sin ( x ) csc ( x )}{csc ( 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 ) - tan ^{2} ( x ) - sin ( x ) csc ( x )}{csc ( x )}

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

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

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

Rewrite using trig identities

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

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

tan^{2} (x)

Use the Pythagorean identity:

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

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

= sec^{2} (x) - 1

Factor

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

sec^{2} (x) - 1

Apply Difference of Two Squares Formula:

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

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

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

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

\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

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

Expand

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

Expand

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

(- 1 + sec (x)) (1 + sec (x))

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = sec (x), b = 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= sec^{2} (x) - 1

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

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

No Solution

sin (x) + sec (x) = 0

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{1 + sin ( x ) cos ( x )}{cos ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

1 + cos (x) sin (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

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

1 + \frac{sin ( 2 x )}{2} = 0

Move

1

to the right side

1 + \frac{sin ( 2 x )}{2} = 0

Subtract

1

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

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

No Solution

sin (x) - sec (x) = 0

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{- 1 + sin ( x ) cos ( x )}{cos ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

- 1 + cos (x) sin (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

Popular Examples

sin(x)=(2*490*10^{-9})/(151*10^{-6)}

sin (x) = \frac{2 \cdot 490 \cdot 1 0 ^{- 9}}{151 \cdot 1 0 ^{- 6}}

cos(x/2)-cos(x)=0

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

cos(2t)=cos(t),0<= t<2pi

cos (2 t) = cos (t), 0 \leq t < 2 π

(cos(2x))^2=cos(x)

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

sin(θ)=sin(-(2pi)/3)

sin (θ) = sin (- \frac{2 π}{3})

Frequently Asked Questions (FAQ)

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