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

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

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tan(a)+cot(a)=sec(a)cos(a)

Solution

tan (a) + cot (a) = sec (a) cos (a)

Solution

No Solution for a \in R

Solution steps

tan (a) + cot (a) = sec (a) cos (a)

Subtract

sec (a) cos (a)

from both sides

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

\frac{cos ( a )}{sin ( a )} + \frac{sin ( a )}{cos ( a )} - cos (a) \frac{1}{cos ( a )} : \frac{cos ^{2} ( a ) + sin ^{2} ( a ) - sin ( a ) cos ( a )}{sin ( a ) cos ( a )}

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (a)

= 1

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

sin (a), cos (a), 1 : sin (a) cos (a)

sin (a), cos (a), 1

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= sin (a) cos (a)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin (a) cos (a)

For

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

multiply the denominator and numerator by

cos (a)

\frac{cos ( a )}{sin ( a )} = \frac{cos ( a ) cos ( a )}{sin ( a ) cos ( a )} = \frac{cos ^{2} ( a )}{sin ( a ) cos ( a )}

For

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

multiply the denominator and numerator by

sin (a)

\frac{sin ( a )}{cos ( a )} = \frac{sin ( a ) sin ( a )}{cos ( a ) sin ( a )} = \frac{sin ^{2} ( a )}{sin ( a ) cos ( a )}

For

\frac{1}{1} :

multiply the denominator and numerator by

sin (a) cos (a)

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

= \frac{cos ^{2} ( a )}{sin ( a ) cos ( a )} + \frac{sin ^{2} ( a )}{sin ( a ) cos ( a )} - \frac{sin ( a ) cos ( a )}{sin ( a ) cos ( a )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ^{2} ( a ) + sin ^{2} ( a ) - sin ( a ) cos ( a )}{sin ( a ) cos ( a )}

= \frac{cos ^{2} ( a ) + sin ^{2} ( a ) - sin ( a ) cos ( a )}{sin ( a ) cos ( a )}

\frac{cos ^{2} ( a ) + sin ^{2} ( a ) - cos ( a ) sin ( a )}{cos ( a ) sin ( a )} = 0

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

cos^{2} (a) + sin^{2} (a) - cos (a) sin (a) = 0

Rewrite using trig identities

- cos (a) sin (a) + 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 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

-5tan^2(θ)+3=8tan(θ)+6

- 5 tan^{2} (θ) + 3 = 8 tan (θ) + 6

cos(θ)=(-3)/(sqrt(13))

cos (θ) = \frac{- 3}{13}

cos(x)=0.84

cos (x) = 0.84

cos(2x)=((4k-5))/(10)

cos (2 x) = \frac{( 4 k - 5 )}{10}

cos^2(2x)-sin^2(2x)=-1

cos^{2} (2 x) - sin^{2} (2 x) = - 1

Frequently Asked Questions (FAQ)

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