What is the general solution for cot(a)sec(a)=cos(a) ?

The general solution for cot(a)sec(a)=cos(a) is No Solution for a\in\mathbb{R}

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Popular >

cot(a)sec(a)=cos(a)

Solution

cot (a) sec (a) = cos (a)

Solution

No Solution for a \in R

Solution steps

cot (a) sec (a) = cos (a)

Subtract

cos (a)

from both sides

cot (a) sec (a) - cos (a) = 0

Express with sin, cos

- cos (a) + cot (a) sec (a)

Use the basic trigonometric identity:

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

= - cos (a) + \frac{cos ( a )}{sin ( a )} sec (a)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (a)

= \frac{1}{sin ( a )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for cot(a)sec(a)=cos(a) ?
The general solution for cot(a)sec(a)=cos(a) is No Solution for a\in\mathbb{R}