What is the general solution for sin(x)= 1/(cos(x)) ?

The general solution for sin(x)= 1/(cos(x)) is No Solution for x\in\mathbb{R}

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sin(x)= 1/(cos(x))

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

No Solution for x \in R

Solution steps

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

Subtract

\frac{1}{cos ( x )}

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

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

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

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

Rewrite using trig identities

sin (x) cos (x) - 1

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 for x \in R

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6 sin (θ) + 3 = 0

2 cos (x) + sin (2 x) = 0

sin (3 θ) = - \frac{1}{2}

sin (x) (2 + 2 cos (x)) = 0

What is the general solution for sin(x)= 1/(cos(x)) ?
The general solution for sin(x)= 1/(cos(x)) is No Solution for x\in\mathbb{R}