What is the general solution for csc(A)*tan(A)=sin(A) ?

The general solution for csc(A)*tan(A)=sin(A) is No Solution for A\in\mathbb{R}

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Popular >

csc(A)*tan(A)=sin(A)

Solution

csc (A) \cdot tan (A) = sin (A)

Solution

No Solution for A \in R

Solution steps

csc (A) tan (A) = sin (A)

Subtract

sin (A)

from both sides

csc (A) tan (A) - sin (A) = 0

Express with sin, cos

- sin (A) + csc (A) tan (A)

Use the basic trigonometric identity:

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

= - sin (A) + \frac{1}{sin ( A )} tan (A)

Use the basic trigonometric identity:

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

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

Simplify

- sin (A) + \frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )} : \frac{- sin ( A ) cos ( A ) + 1}{cos ( A )}

- sin (A) + \frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )}

\frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )} = \frac{1}{cos ( A )}

\frac{1}{sin ( A )} \cdot \frac{sin ( A )}{cos ( A )}

Multiply fractions:

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

= \frac{1 \cdot sin ( A )}{sin ( A ) cos ( A )}

Cancel the common factor:

sin (A)

= \frac{1}{cos ( A )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

1 - cos (A) sin (A) = 0

Rewrite using trig identities

1 - cos (A) sin (A)

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 A )}{2}

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

- sin (2 A) = - 2

- sin (2 A) = - 2

Divide both sides by

- 1

- sin (2 A) = - 2

Divide both sides by

- 1

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

Simplify

sin (2 A) = 2

sin (2 A) = 2

- 1 \leq sin (x) \leq 1

No Solution for A \in R

Popular Examples

cos(x)=sqrt(2/3)

cos (x) = \frac{2}{3}

(sec(3x)-sqrt(2))(tan(2x)-1)=0

(sec (3 x) - 2) (tan (2 x) - 1) = 0

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

cos (2 x) \cdot 2 - 2 cos (x) = 0

-2sin(2x)= pi/(12)

- 2 sin (2 x) = \frac{π}{12}

cos(x)=(sec(x))/4

cos (x) = \frac{sec ( x )}{4}

Frequently Asked Questions (FAQ)

What is the general solution for csc(A)*tan(A)=sin(A) ?
The general solution for csc(A)*tan(A)=sin(A) is No Solution for A\in\mathbb{R}