What is the general solution for sin(x)= 4/(cos(x)) ?

The general solution for sin(x)= 4/(cos(x)) is No Solution for x\in\mathbb{R}

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sin(x)= 4/(cos(x))

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

No Solution for x \in R

Solution steps

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

Subtract

\frac{4}{cos ( x )}

from both sides

sin (x) - \frac{4}{cos ( x )} = 0

Simplify

sin (x) - \frac{4}{cos ( x )} : \frac{sin ( x ) cos ( x ) - 4}{cos ( x )}

sin (x) - \frac{4}{cos ( x )}

Convert element to fraction:

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

= \frac{sin ( x ) cos ( x )}{cos ( x )} - \frac{4}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) cos ( x ) - 4}{cos ( x )}

\frac{sin ( x ) cos ( x ) - 4}{cos ( x )} = 0

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

sin (x) cos (x) - 4 = 0

Rewrite using trig identities

sin (x) cos (x) - 4

Use the Double Angle identity:

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

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

= - 4 + \frac{sin ( 2 x )}{2}

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

Move

4

to the right side

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

Add

4

to both sides

- 4 + \frac{sin ( 2 x )}{2} + 4 = 0 + 4

Simplify

\frac{sin ( 2 x )}{2} = 4

\frac{sin ( 2 x )}{2} = 4

Multiply both sides by

2

\frac{sin ( 2 x )}{2} = 4

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} = 4 \cdot 2

Simplify

sin (2 x) = 8

sin (2 x) = 8

- 1 \leq sin (x) \leq 1

No Solution for x \in R

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2 csc (θ) + 2 = 0

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What is the general solution for sin(x)= 4/(cos(x)) ?
The general solution for sin(x)= 4/(cos(x)) is No Solution for x\in\mathbb{R}