What is the general solution for sec(x)tan(x)-cos(x)cot(x)=sin(x) ?

The general solution for sec(x)tan(x)-cos(x)cot(x)=sin(x) is x= pi/4+pin,x=(3pi)/4+pin

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sec(x)tan(x)-cos(x)cot(x)=sin(x)

Solution

sec (x) tan (x) - cos (x) cot (x) = sin (x)

Solution

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

sec (x) tan (x) - cos (x) cot (x) = sin (x)

Subtract

sin (x)

from both sides

sec (x) tan (x) - cos (x) cot (x) - sin (x) = 0

Express with sin, cos

- sin (x) - cos (x) cot (x) + sec (x) tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

Convert element to fraction:

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

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

Least Common Multiplier of

1, sin (x), cos^{2} (x) : cos^{2} (x) sin (x)

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

Lowest Common Multiplier (LCM)

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

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

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

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

For

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

multiply the denominator and numerator by

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

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

For

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

multiply the denominator and numerator by

cos^{2} (x)

\frac{cos ^{2} ( x )}{sin ( x )} = \frac{cos ^{2} ( x ) cos ^{2} ( x )}{sin ( x ) cos ^{2} ( x )} = \frac{cos ^{4} ( x )}{cos ^{2} ( x ) sin ( x )}

For

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

multiply the denominator and numerator by

sin (x)

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

= - \frac{cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )} - \frac{cos ^{4} ( x )}{cos ^{2} ( x ) sin ( x )} + \frac{sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- cos ^{2} ( x ) sin ^{2} ( x ) - cos ^{4} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )}

= \frac{- cos ^{2} ( x ) sin ^{2} ( x ) - cos ^{4} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )}

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - sin^{2} (x), b = 1, c = sin^{2} (x)

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

- 1 \cdot sin^{2} (x) + sin^{2} (x) sin^{2} (x)

1 \cdot sin^{2} (x) = sin^{2} (x)

1 \cdot sin^{2} (x)

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

= sin^{2} (x)

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

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

Apply exponent rule:

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

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

= sin^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= sin^{4} (x)

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

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

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

Add similar elements:

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

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

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

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

Factor

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

- cos^{4} (x) + sin^{4} (x)

Rewrite

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

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

= (sin (x) + cos (x)) (sin (x) - cos (x))

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= (- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1

Simplify

(- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1 : (- cos (x) + sin (x)) (cos (x) + sin (x))

(- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1

Multiply:

(cos (x) + sin (x)) \cdot 1 = (cos (x) + sin (x))

= (sin (x) - cos (x)) (cos (x) + sin (x))

= (- cos (x) + sin (x)) (cos (x) + sin (x))

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

Solving each part separately

- cos (x) + sin (x) = 0 or cos (x) + sin (x) = 0

- cos (x) + sin (x) = 0 : x = \frac{π}{4} + πn

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

Rewrite using trig identities

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

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

- 1 + tan (x) = 0

- 1 + tan (x) = 0

Move

1

to the right side

- 1 + tan (x) = 0

Add

1

to both sides

- 1 + tan (x) + 1 = 0 + 1

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

cos (x) + sin (x) = 0

Rewrite using trig identities

cos (x) + sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{cos ( x ) + sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

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

Use the basic trigonometric identity:

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

1 + tan (x) = 0

1 + tan (x) = 0

Move

1

to the right side

1 + tan (x) = 0

Subtract

1

from both sides

1 + tan (x) - 1 = 0 - 1

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Popular Examples

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

sin(2x)=-5cos(2x)

sin (2 x) = - 5 cos (2 x)

tan(x)cot(x)=sec(x)csc(x)

tan (x) cot (x) = sec (x) csc (x)

4sin(x)=cos(x)-2

4 sin (x) = cos (x) - 2

sin(θ)=1-cos(θ)

sin (θ) = 1 - cos (θ)

Frequently Asked Questions (FAQ)

What is the general solution for sec(x)tan(x)-cos(x)cot(x)=sin(x) ?
The general solution for sec(x)tan(x)-cos(x)cot(x)=sin(x) is x= pi/4+pin,x=(3pi)/4+pin