What is the general solution for csc(2θ)= 1/2 (sec(θ))(cos(θ)) ?

The general solution for csc(2θ)= 1/2 (sec(θ))(cos(θ)) is No Solution for θ\in\mathbb{R}

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csc(2θ)= 1/2 (sec(θ))(cos(θ))

csc (2 θ) = \frac{1}{2} (sec (θ)) (cos (θ))

No Solution for θ \in R

Solution steps

csc (2 θ) = \frac{1}{2} (sec (θ)) (cos (θ))

Subtract

\frac{1}{2} sec (θ) cos (θ)

from both sides

csc (2 θ) - \frac{1}{2} sec (θ) cos (θ) = 0

Simplify

csc (2 θ) - \frac{1}{2} sec (θ) cos (θ) : \frac{2 csc ( 2 θ ) - sec ( θ ) cos ( θ )}{2}

csc (2 θ) - \frac{1}{2} sec (θ) cos (θ)

\frac{1}{2} sec (θ) cos (θ) = \frac{sec ( θ ) cos ( θ )}{2}

\frac{1}{2} sec (θ) cos (θ)

Multiply fractions:

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

= \frac{1 \cdot sec ( θ ) cos ( θ )}{2}

Multiply:

1 \cdot sec (θ) = sec (θ)

= \frac{sec ( θ ) cos ( θ )}{2}

= csc (2 θ) - \frac{sec ( θ ) cos ( θ )}{2}

Convert element to fraction:

csc (2 θ) = \frac{csc ( 2 θ ) 2}{2}

= \frac{csc ( 2 θ ) \cdot 2}{2} - \frac{sec ( θ ) cos ( θ )}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{csc ( 2 θ ) \cdot 2 - sec ( θ ) cos ( θ )}{2}

\frac{2 csc ( 2 θ ) - sec ( θ ) cos ( θ )}{2} = 0

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

2 csc (2 θ) - sec (θ) cos (θ) = 0

Rewrite using trig identities

2 csc (2 θ) - sec (θ) cos (θ)

sec (θ) cos (θ) = 1

sec (θ) cos (θ)

Express with sin, cos

sec (θ) cos (θ)

Use the basic trigonometric identity:

sec (θ) = \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )} cos (θ)

\frac{1}{cos ( θ )} cos (θ) = 1

\frac{1}{cos ( θ )} cos (θ)

Multiply fractions:

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

= \frac{1 cos ( θ )}{cos ( θ )}

Cancel the common factor:

cos (θ)

= 1

= 1

= 2 csc (2 θ) - 1

2 csc (2 θ) - 1 = 0

Move

1

to the right side

2 csc (2 θ) - 1 = 0

Add

1

to both sides

2 csc (2 θ) - 1 + 1 = 0 + 1

Simplify

2 csc (2 θ) = 1

2 csc (2 θ) = 1

Divide both sides by

2

2 csc (2 θ) = 1

Divide both sides by

2

\frac{2 csc ( 2 θ )}{2} = \frac{1}{2}

Simplify

csc (2 θ) = \frac{1}{2}

csc (2 θ) = \frac{1}{2}

csc (x) \leq - 1 or csc (x) \geq 1

No Solution for θ \in R

sec^{2} (x) + 2 sec (x) - 8 = 0

3 cos^{2} (x) + 4 sin (x) - 4 = 0

sin (x) = \frac{7}{15}

3 sec (x) = 9

cos^{2} (x) + 2 cos (x) + 1 = 0, 0 \leq x \leq 2 π

What is the general solution for csc(2θ)= 1/2 (sec(θ))(cos(θ)) ?
The general solution for csc(2θ)= 1/2 (sec(θ))(cos(θ)) is No Solution for θ\in\mathbb{R}