What is the general solution for tan(θ)sec(θ)=cot(θ)csc(θ) ?

The general solution for tan(θ)sec(θ)=cot(θ)csc(θ) is θ= pi/4+pin

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tan(θ)sec(θ)=cot(θ)csc(θ)

Solution

tan (θ) sec (θ) = cot (θ) csc (θ)

Solution

θ = \frac{π}{4} + πn

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

tan (θ) sec (θ) = cot (θ) csc (θ)

Subtract

cot (θ) csc (θ)

from both sides

tan (θ) sec (θ) - cot (θ) csc (θ) = 0

Express with sin, cos

- cot (θ) csc (θ) + sec (θ) tan (θ)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

= - \frac{cos ( θ )}{sin ( θ )} \cdot \frac{1}{sin ( θ )} + sec (θ) tan (θ)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

- \frac{cos ( θ )}{sin ( θ )} \cdot \frac{1}{sin ( θ )} + \frac{1}{cos ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )} : \frac{- cos ^{3} ( θ ) + sin ^{3} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

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

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

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

Multiply fractions:

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

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

Multiply:

cos (θ) \cdot 1 = cos (θ)

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

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Apply exponent rule:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= sin^{2} (θ)

= \frac{cos ( θ )}{sin ^{2} ( θ )}

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot sin (θ) = sin (θ)

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

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

Apply exponent rule:

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

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= cos^{2} (θ)

= \frac{sin ( θ )}{cos ^{2} ( θ )}

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

Least Common Multiplier of

sin^{2} (θ), cos^{2} (θ) : sin^{2} (θ) cos^{2} (θ)

sin^{2} (θ), cos^{2} (θ)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin^{2} (θ)

cos^{2} (θ)

= sin^{2} (θ) cos^{2} (θ)

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^{2} (θ) cos^{2} (θ)

For

\frac{cos ( θ )}{sin ^{2} ( θ )} :

multiply the denominator and numerator by

cos^{2} (θ)

\frac{cos ( θ )}{sin ^{2} ( θ )} = \frac{cos ( θ ) cos ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )} = \frac{cos ^{3} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

For

\frac{sin ( θ )}{cos ^{2} ( θ )} :

multiply the denominator and numerator by

sin^{2} (θ)

\frac{sin ( θ )}{cos ^{2} ( θ )} = \frac{sin ( θ ) sin ^{2} ( θ )}{cos ^{2} ( θ ) sin ^{2} ( θ )} = \frac{sin ^{3} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

= - \frac{cos ^{3} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )} + \frac{sin ^{3} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{- cos ^{3} ( θ ) + sin ^{3} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

= \frac{- cos ^{3} ( θ ) + sin ^{3} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

\frac{- cos ^{3} ( θ ) + sin ^{3} ( θ )}{cos ^{2} ( θ ) sin ^{2} ( θ )} = 0

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

- cos^{3} (θ) + sin^{3} (θ) = 0

Factor

- cos^{3} (θ) + sin^{3} (θ) : (sin (θ) - cos (θ)) (sin^{2} (θ) + sin (θ) cos (θ) + cos^{2} (θ))

- cos^{3} (θ) + sin^{3} (θ)

Apply Difference of Cubes Formula:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

sin^{3} (θ) - cos^{3} (θ) = (sin (θ) - cos (θ)) (sin^{2} (θ) + sin (θ) cos (θ) + cos^{2} (θ))

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= (- cos (θ) + sin (θ)) (cos (θ) sin (θ) + 1)

(- cos (θ) + sin (θ)) (cos (θ) sin (θ) + 1) = 0

Solving each part separately

- cos (θ) + sin (θ) = 0 or cos (θ) sin (θ) + 1 = 0

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

- cos (θ) + sin (θ) = 0

Rewrite using trig identities

- cos (θ) + sin (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

- 1 + \frac{sin ( θ )}{cos ( θ )} = 0

Use the basic trigonometric identity:

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

- 1 + tan (θ) = 0

- 1 + tan (θ) = 0

Move

1

to the right side

- 1 + tan (θ) = 0

Add

1

to both sides

- 1 + tan (θ) + 1 = 0 + 1

Simplify

tan (θ) = 1

tan (θ) = 1

General solutions for

tan (θ) = 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}

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

cos (θ) sin (θ) + 1 = 0 :

No Solution

cos (θ) sin (θ) + 1 = 0

Rewrite using trig identities

cos (θ) sin (θ) + 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 θ )}{2}

1 + \frac{sin ( 2 θ )}{2} = 0

Move

1

to the right side

1 + \frac{sin ( 2 θ )}{2} = 0

Subtract

1

from both sides

1 + \frac{sin ( 2 θ )}{2} - 1 = 0 - 1

Simplify

\frac{sin ( 2 θ )}{2} = - 1

\frac{sin ( 2 θ )}{2} = - 1

Multiply both sides by

2

\frac{sin ( 2 θ )}{2} = - 1

Multiply both sides by

2

\frac{2 sin ( 2 θ )}{2} = 2 (- 1)

Simplify

sin (2 θ) = - 2

sin (2 θ) = - 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

θ = \frac{π}{4} + πn

Popular Examples

0=4+3cos(x)

0 = 4 + 3 cos (x)

cos^2(x)-cos(x)=0.75

cos^{2} (x) - cos (x) = 0.75

2sin^3(x)+cos^2(x)=1

2 sin^{3} (x) + cos^{2} (x) = 1

2tan(x)=tan(x)tan(x)

2 tan (x) = tan (x) tan (x)

(2tan(x))/(1-tan^2(x))=sqrt(3)

\frac{2 tan ( x )}{1 - tan ^{2} ( x )} = 3

Frequently Asked Questions (FAQ)

What is the general solution for tan(θ)sec(θ)=cot(θ)csc(θ) ?
The general solution for tan(θ)sec(θ)=cot(θ)csc(θ) is θ= pi/4+pin