What is the general solution for (sin(x))/(cos(x))+(cos(x))/(sec(x))=1 ?

The general solution for (sin(x))/(cos(x))+(cos(x))/(sec(x))=1 is x=2pin,x=pi+2pin

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(sin(x))/(cos(x))+(cos(x))/(sec(x))=1

Solution

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

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

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

Simplify

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

cos (x), sec (x), 1 : cos (x) sec (x)

cos (x), sec (x), 1

Lowest Common Multiplier (LCM)

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

= cos (x) sec (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 (x) sec (x)

For

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

multiply the denominator and numerator by

sec (x)

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

For

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

multiply the denominator and numerator by

cos (x)

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

For

\frac{1}{1} :

multiply the denominator and numerator by

cos (x) sec (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

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

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

cos^{2} (x)

Use the Pythagorean identity:

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

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

= 1 - sin^{2} (x)

Factor

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

1 - sin^{2} (x)

Apply Difference of Two Squares Formula:

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

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

= (1 + sin (x)) (1 - sin (x))

= (1 + sin (x)) (1 - sin (x))

= (1 + sin (x)) (1 - sin (x)) - cos (x) sec (x) + sec (x) sin (x)

Use the basic trigonometric identity:

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

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

Simplify

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

(1 + sin (x)) (1 - sin (x)) - \frac{1}{sec ( x )} sec (x) + sec (x) sin (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

sec (x)

= 1

= (sin (x) + 1) (- sin (x) + 1) - 1 + sec (x) sin (x)

Expand

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

(1 + sin (x)) (1 - sin (x))

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = sin (x)

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (x)

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

Simplify

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

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

Group like terms

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

1 - 1 = 0

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

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

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

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

Factor

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

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

Apply exponent rule:

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

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

= - sin (x) sin (x) + sec (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- sin (x) + sec (x))

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

- sin (x) + sec (x) = 0 :

No Solution

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

Express with sin, cos

sec (x) - sin (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

1 - cos (x) sin (x)

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 x )}{2}

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

- sin (2 x) = - 2

- sin (2 x) = - 2

Divide both sides by

- 1

- sin (2 x) = - 2

Divide both sides by

- 1

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

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (sin(x))/(cos(x))+(cos(x))/(sec(x))=1 ?
The general solution for (sin(x))/(cos(x))+(cos(x))/(sec(x))=1 is x=2pin,x=pi+2pin