solvefor x,sin(x)+cos(y)=sin(x)cos(y)

Solution

solve for

x, sin (x) + cos (y) = sin (x) cos (y)

x = arccsc (\frac{cos ( y ) - 1}{cos ( y )}) + 2 πn, x = π + arccsc (- \frac{cos ( y ) - 1}{cos ( y )}) + 2 πn

Solution steps

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

Subtract

sin (x) cos (y)

from both sides

sin (x) + cos (y) - cos (y) sin (x) = 0

Rewrite using trig identities

cos (y) + \frac{1 - cos ( y )}{csc ( x )} = 0

Multiply both sides by

csc (x)

cos (y) csc (x) + 1 - cos (y) = 0

Move

cos (y)

to the right side

cos (y) csc (x) + 1 = cos (y)

Move

1

to the right side

cos (y) csc (x) = cos (y) - 1

Divide both sides by

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

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

Apply trig inverse properties

x = arccsc (\frac{cos ( y ) - 1}{cos ( y )}) + 2 πn, x = π + arccsc (- \frac{cos ( y ) - 1}{cos ( y )}) + 2 πn

x, z = arctan (\frac{x}{y})

- \frac{1}{2} = cos (θ)

4 cos^{2} (x) - 3 cos (x) - 2 = 0

sin (x) = cos (3 x)

x, y = 5 sin (\frac{x}{4})