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solvefor y,-ln(cos(x))+cos(x)sin(y)=C

Solution

solve for

y, - ln (cos (x)) + cos (x) sin (y) = C

Solution

y = arcsin (\frac{C + ln ( cos ( x ) )}{cos ( x )}) + 2 πn, y = π + arcsin (- \frac{C + ln ( cos ( x ) )}{cos ( x )}) + 2 πn

Solution steps

- ln (cos (x)) + cos (x) sin (y) = C

Move

ln (cos (x))

to the right side

- ln (cos (x)) + cos (x) sin (y) = C

Add

ln (cos (x))

to both sides

- ln (cos (x)) + cos (x) sin (y) + ln (cos (x)) = C + ln (cos (x))

Simplify

cos (x) sin (y) = C + ln (cos (x))

cos (x) sin (y) = C + ln (cos (x))

Divide both sides by

cos (x); x \neq = \frac{π}{2} + 2 πn, x \neq = \frac{3 π}{2} + 2 πn

cos (x) sin (y) = C + ln (cos (x))

Divide both sides by

cos (x); x \neq = \frac{π}{2} + 2 πn, x \neq = \frac{3 π}{2} + 2 πn

\frac{cos ( x ) sin ( y )}{cos ( x )} = \frac{C}{cos ( x )} + \frac{ln ( cos ( x ) )}{cos ( x )}; x \neq = \frac{π}{2} + 2 πn, x \neq = \frac{3 π}{2} + 2 πn

Simplify

\frac{cos ( x ) sin ( y )}{cos ( x )} = \frac{C}{cos ( x )} + \frac{ln ( cos ( x ) )}{cos ( x )}

Simplify

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

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

Cancel the common factor:

cos (x)

= sin (y)

Simplify

\frac{C}{cos ( x )} + \frac{ln ( cos ( x ) )}{cos ( x )} : \frac{C + ln ( cos ( x ) )}{cos ( x )}

\frac{C}{cos ( x )} + \frac{ln ( cos ( x ) )}{cos ( x )}

Apply rule

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

= \frac{C + ln ( cos ( x ) )}{cos ( x )}

sin (y) = \frac{C + ln ( cos ( x ) )}{cos ( x )}; x \neq = \frac{π}{2} + 2 πn, x \neq = \frac{3 π}{2} + 2 πn

sin (y) = \frac{C + ln ( cos ( x ) )}{cos ( x )}; x \neq = \frac{π}{2} + 2 πn, x \neq = \frac{3 π}{2} + 2 πn

sin (y) = \frac{C + ln ( cos ( x ) )}{cos ( x )}; x \neq = \frac{π}{2} + 2 πn, x \neq = \frac{3 π}{2} + 2 πn

Apply trig inverse properties

sin (y) = \frac{C + ln ( cos ( x ) )}{cos ( x )}

General solutions for

sin (y) = \frac{C + ln ( cos ( x ) )}{cos ( x )}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

y = arcsin (\frac{C + ln ( cos ( x ) )}{cos ( x )}) + 2 πn, y = π + arcsin (- \frac{C + ln ( cos ( x ) )}{cos ( x )}) + 2 πn

y = arcsin (\frac{C + ln ( cos ( x ) )}{cos ( x )}) + 2 πn, y = π + arcsin (- \frac{C + ln ( cos ( x ) )}{cos ( x )}) + 2 πn

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