What is the general solution for 1/(1-sin(x))=2sec^2(x)+tan(x)sec(x) ?

The general solution for 1/(1-sin(x))=2sec^2(x)+tan(x)sec(x) is No Solution for x\in\mathbb{R}

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1/(1-sin(x))=2sec^2(x)+tan(x)sec(x)

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

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

from both sides

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

Simplify

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

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

Convert element to fraction:

2 sec^{2} (x) = \frac{2 sec ^{2} ( x ) ( 1 - sin ( x ) )}{1 - sin ( x )}, tan (x) sec (x) = \frac{tan ( x ) sec ( x ) ( 1 - sin ( x ) )}{1 - sin ( x )}

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

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

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

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

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

Multiply fractions:

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

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

Multiply:

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

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

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

Combine the fractions

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

Apply rule

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

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

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

Expand

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

- 2 (1 - sin (x)) - sin (x) (1 - sin (x))

Expand

- 2 (1 - sin (x)) : - 2 + 2 sin (x)

- 2 (1 - sin (x))

Apply the distributive law:

a (b - c) = ab - a c

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

= - 2 \cdot 1 - (- 2) sin (x)

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 1 + 2 sin (x)

Multiply the numbers:

2 \cdot 1 = 2

= - 2 + 2 sin (x)

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - sin (x), b = 1, c = sin (x)

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

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

1 \cdot sin (x)

Multiply:

1 \cdot 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)

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

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

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

Add similar elements:

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= - 2 + sin (x) + 1

- 2 + sin (x) + 1 = 0

Group like terms

sin (x) - 2 + 1 = 0

Add/Subtract the numbers:

- 2 + 1 = - 1

sin (x) - 1 = 0

Move

1

to the right side

sin (x) - 1 = 0

Add

1

to both sides

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

Simplify

sin (x) = 1

sin (x) = 1

General solutions for

sin (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 1/(1-sin(x))=2sec^2(x)+tan(x)sec(x) ?
The general solution for 1/(1-sin(x))=2sec^2(x)+tan(x)sec(x) is No Solution for x\in\mathbb{R}