What is the general solution for sin(x)-2cos(x)cot(x)=csc(x) ?

The general solution for sin(x)-2cos(x)cot(x)=csc(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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sin(x)-2cos(x)cot(x)=csc(x)

Solution

sin (x) - 2 cos (x) cot (x) = csc (x)

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) - 2 cos (x) cot (x) = csc (x)

Subtract

csc (x)

from both sides

sin (x) - 2 cos (x) cot (x) - csc (x) = 0

Express with sin, cos

sin (x) - 2 cos (x) \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )} = 0

Simplify

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

sin (x) - 2 cos (x) \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )}

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

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

Multiply fractions:

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

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

cos (x) \cdot 2 cos (x) = 2 cos^{2} (x)

cos (x) \cdot 2 cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= 2 cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= 2 cos^{2} (x)

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

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

Combine the fractions

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

Apply rule

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

\frac{sin ^{2} ( x ) - 2 cos ^{2} ( x ) - 1}{sin ( x )} = 0

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

sin^{2} (x) - 2 cos^{2} (x) - 1 = 0

Add

2 cos^{2} (x)

to both sides

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

Square both sides

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

Subtract

(2 cos^{2} (x))^{2}

from both sides

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x) = 0

Factor

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x) : (sin^{2} (x) - 1 + 2 cos^{2} (x)) (sin^{2} (x) - 1 - 2 cos^{2} (x))

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x)

Rewrite

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x)

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

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x)

Rewrite

4

2^{2}

= (sin^{2} (x) - 1)^{2} - 2^{2} cos^{4} (x)

Apply exponent rule:

a^{b c} = (a^{b})^{c}

cos^{4} (x) = (cos^{2} (x))^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

2^{2} (cos^{2} (x))^{2} = (2 cos^{2} (x))^{2}

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

Refine

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

(sin^{2} (x) - 1 + 2 cos^{2} (x)) (sin^{2} (x) - 1 - 2 cos^{2} (x)) = 0

Solving each part separately

sin^{2} (x) - 1 + 2 cos^{2} (x) = 0 or sin^{2} (x) - 1 - 2 cos^{2} (x) = 0

sin^{2} (x) - 1 + 2 cos^{2} (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin^{2} (x) - 1 + 2 cos^{2} (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

= cos^{2} (x)

cos^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin^{2} (x) - 1 - 2 cos^{2} (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin^{2} (x) - 1 - 2 cos^{2} (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

= - 3 cos^{2} (x)

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

Divide both sides by

- 3

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

Divide both sides by

- 3

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

Divide both sides by

- 3

\frac{- 3 cos ^{2} ( x )}{- 3} = \frac{0}{- 3}

Simplify

cos^{2} (x) = 0

cos^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sin (x) - 2 cos (x) cot (x) = csc (x)

Remove the ones that don't agree with the equation.

Check the solution

\frac{π}{2} + 2 πn :

True

\frac{π}{2} + 2 πn

Plug in

n = 1

\frac{π}{2} + 2 π 1

For

sin (x) - 2 cos (x) cot (x) = csc (x)

plug in

x = \frac{π}{2} + 2 π 1

sin (\frac{π}{2} + 2 π 1) - 2 cos (\frac{π}{2} + 2 π 1) cot (\frac{π}{2} + 2 π 1) = csc (\frac{π}{2} + 2 π 1)

Refine

1 = 1

\Rightarrow True

Check the solution

\frac{3 π}{2} + 2 πn :

True

\frac{3 π}{2} + 2 πn

Plug in

n = 1

\frac{3 π}{2} + 2 π 1

For

sin (x) - 2 cos (x) cot (x) = csc (x)

plug in

x = \frac{3 π}{2} + 2 π 1

sin (\frac{3 π}{2} + 2 π 1) - 2 cos (\frac{3 π}{2} + 2 π 1) cot (\frac{3 π}{2} + 2 π 1) = csc (\frac{3 π}{2} + 2 π 1)

Refine

- 1 = - 1

\Rightarrow True

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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tan (18 0^{\circ} - θ) = 33

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)-2cos(x)cot(x)=csc(x) ?
The general solution for sin(x)-2cos(x)cot(x)=csc(x) is x= pi/2+2pin,x=(3pi)/2+2pin