What is the general solution for solvefor x,sin(x)=cos(y) ?

The general solution for solvefor x,sin(x)=cos(y) is x= pi/2-y+2pin,x=pi-pi/2+y+2pin

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

solve for

x, sin (x) = cos (y)

x = \frac{π}{2} - y + 2 πn, x = π - \frac{π}{2} + y + 2 πn

Solution steps

sin (x) = cos (y)

Rewrite using trig identities

cos (y)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (\frac{π}{2} - y)

sin (x) = sin (\frac{π}{2} - y)

Apply trig inverse properties

sin (x) = sin (\frac{π}{2} - y)

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

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

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

x = \frac{π}{2} - y + 2 πn : x = \frac{π}{2} - y + 2 πn

x = \frac{π}{2} - y + 2 πn

Could not simplify further

x = \frac{π}{2} - y + 2 πn

x = π - (\frac{π}{2} - y) + 2 πn : x = π - \frac{π}{2} + y + 2 πn

x = π - (\frac{π}{2} - y) + 2 πn

- (\frac{π}{2} - y) : - \frac{π}{2} + y

- (\frac{π}{2} - y)

Distribute parentheses

= - \frac{π}{2} - (- y)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - \frac{π}{2} + y

x = π - \frac{π}{2} + y + 2 πn

x = \frac{π}{2} - y + 2 πn, x = π - \frac{π}{2} + y + 2 πn

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

2 cos (x) + 2 sin (x) = 6

6 tan (θ) + 1 = 0

cot (5 x) = - 1

2 cot (θ) + csc^{2} (θ) = 0

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