What is the general solution for tan(x)+cot(x)=2sec(x) ?

The general solution for tan(x)+cot(x)=2sec(x) is x= pi/6+2pin,x=(5pi)/6+2pin

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tan(x)+cot(x)=2sec(x)

Solution

tan (x) + cot (x) = 2 sec (x)

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (x) + cot (x) = 2 sec (x)

Subtract

2 sec (x)

from both sides

tan (x) + cot (x) - 2 sec (x) = 0

Express with sin, cos

cot (x) + tan (x) - 2 sec (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Combine the fractions

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

Apply rule

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

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

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

Least Common Multiplier of

sin (x), cos (x) : sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= sin (x) cos (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

sin (x) cos (x)

For

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

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

sin (x)

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

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

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

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

sin (x) (- 2 + sin (x))

Apply the distributive law:

a (b + c) = ab + a c

a = sin (x), b = - 2, c = sin (x)

= sin (x) (- 2) + sin (x) sin (x)

Apply minus-plus rules

+ (- a) = - a

= - 2 sin (x) + sin (x) sin (x)

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)

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

= - 2 sin (x) + 1

= - 2 sin (x) + 1

= - 2 sin (x) + 1

1 - 2 sin (x) = 0

Move

1

to the right side

1 - 2 sin (x) = 0

Subtract

1

from both sides

1 - 2 sin (x) - 1 = 0 - 1

Simplify

- 2 sin (x) = - 1

- 2 sin (x) = - 1

Divide both sides by

- 2

- 2 sin (x) = - 1

Divide both sides by

- 2

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

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Popular Examples

12sin^2(θ)=9

12 sin^{2} (θ) = 9

2cos(x)+sqrt(3)=0,0<= x<= 2pi

2 cos (x) + 3 = 0, 0 \leq x \leq 2 π

(cot^2(x))/(1+sin(x))=(csc(x)-1)/(sin^2(x))

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

tan(x)=2.3

tan (x) = 2.3

2sec^2(x)=5tan(x)+5

2 sec^{2} (x) = 5 tan (x) + 5

Frequently Asked Questions (FAQ)

What is the general solution for tan(x)+cot(x)=2sec(x) ?
The general solution for tan(x)+cot(x)=2sec(x) is x= pi/6+2pin,x=(5pi)/6+2pin