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Popular >

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

Solution

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

Solution

x = - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) - πn

Solution steps

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

Divide both sides by

sec (a) (1 + sin (- a))

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

Divide both sides by

sec (a) (1 + sin (- a))

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

Simplify

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

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

Move

sec (a)

to the right side

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

Subtract

sec (a)

from both sides

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

Simplify

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

\frac{- tan ( - x )}{- 1} = \frac{\frac{1}{s e c ( a ) ( 1 + s i n ( - a ) )}}{- 1} - \frac{sec ( a )}{- 1}

Simplify

\frac{- tan ( - x )}{- 1} = \frac{\frac{1}{s e c ( a ) ( 1 + s i n ( - a ) )}}{- 1} - \frac{sec ( a )}{- 1}

Simplify

\frac{- tan ( - x )}{- 1} : tan (- x)

\frac{- tan ( - x )}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{tan ( - x )}{1}

Apply rule

\frac{a}{1} = a

= tan (- x)

Simplify

\frac{\frac{1}{s e c ( a ) ( 1 + s i n ( - a ) )}}{- 1} - \frac{sec ( a )}{- 1} : - \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}

\frac{\frac{1}{s e c ( a ) ( 1 + s i n ( - a ) )}}{- 1} - \frac{sec ( a )}{- 1}

Apply rule

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

= \frac{\frac{1}{s e c ( a ) ( s i n ( - a ) + 1 )} - sec ( a )}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{\frac{1}{s e c ( a ) ( 1 + s i n ( - a ) )} - sec ( a )}{1}

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

1 - sec (a) sec (a) (1 + sin (- a))

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

sec (a) sec (a) (1 + sin (- a))

Apply exponent rule:

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

sec (a) sec (a) = sec^{1 + 1} (a)

= sec^{1 + 1} (a) (1 + sin (- a))

Add the numbers:

1 + 1 = 2

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

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

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

= - \frac{\frac{- s e c ^{2} ( a ) ( s i n ( - a ) + 1 ) + 1}{s e c ( a ) ( s i n ( - a ) + 1 )}}{1}

Apply the fraction rule:

\frac{a}{1} = a

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

tan (x) = a \Rightarrow x = arctan (a) + πn

- x = arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) + πn

- x = arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) + πn

Solve

- x = arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) + πn : x = - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) - πn

- x = arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) + πn

Divide both sides by

- 1

- x = arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) + πn

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{arctan ( - \frac{1 - s e c ^{2} ( a ) ( 1 + s i n ( - a ) )}{s e c ( a ) ( 1 + s i n ( - a ) )} )}{- 1} + \frac{πn}{- 1}

Simplify

\frac{- x}{- 1} = \frac{arctan ( - \frac{1 - s e c ^{2} ( a ) ( 1 + s i n ( - a ) )}{s e c ( a ) ( 1 + s i n ( - a ) )} )}{- 1} + \frac{πn}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{arctan ( - \frac{1 - s e c ^{2} ( a ) ( 1 + s i n ( - a ) )}{s e c ( a ) ( 1 + s i n ( - a ) )} )}{- 1} + \frac{πn}{- 1} : - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) - πn

\frac{arctan ( - \frac{1 - s e c ^{2} ( a ) ( 1 + s i n ( - a ) )}{s e c ( a ) ( 1 + s i n ( - a ) )} )}{- 1} + \frac{πn}{- 1}

\frac{arctan ( - \frac{1 - s e c ^{2} ( a ) ( 1 + s i n ( - a ) )}{s e c ( a ) ( 1 + s i n ( - a ) )} )}{- 1} = - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )})

\frac{arctan ( - \frac{1 - s e c ^{2} ( a ) ( 1 + s i n ( - a ) )}{s e c ( a ) ( 1 + s i n ( - a ) )} )}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{arctan ( - \frac{1 - s e c ^{2} ( a ) ( 1 + s i n ( - a ) )}{s e c ( a ) ( 1 + s i n ( - a ) )} )}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{arctan ( - \frac{1 - s e c ^{2} ( a ) ( 1 + s i n ( - a ) )}{s e c ( a ) ( 1 + s i n ( - a ) )} )}{1} = arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )})

= - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )})

= - arctan (- \frac{- sec ^{2} ( a ) ( sin ( - a ) + 1 ) + 1}{sec ( a ) ( sin ( - a ) + 1 )}) + \frac{πn}{- 1}

\frac{πn}{- 1} = - πn

\frac{πn}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{πn}{1}

Apply rule

\frac{a}{1} = a

= - πn

= - arctan (- \frac{- sec ^{2} ( a ) ( sin ( - a ) + 1 ) + 1}{sec ( a ) ( sin ( - a ) + 1 )}) - πn

x = - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) - πn

x = - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) - πn

x = - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) - πn

x = - arctan (- \frac{1 - sec ^{2} ( a ) ( 1 + sin ( - a ) )}{sec ( a ) ( 1 + sin ( - a ) )}) - πn

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