What is the general solution for sin(x)+pi/2 =cos(x) ?

The general solution for sin(x)+pi/2 =cos(x) is No Solution for x\in\mathbb{R}

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sin(x)+pi/2 =cos(x)

sin (x) + \frac{π}{2} = cos (x)

No Solution for x \in R

Solution steps

sin (x) + \frac{π}{2} = cos (x)

Square both sides

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

Subtract

cos^{2} (x)

from both sides

sin^{2} (x) + π sin (x) + \frac{π ^{2}}{4} - cos^{2} (x) = 0

Simplify

sin^{2} (x) + π sin (x) + \frac{π ^{2}}{4} - cos^{2} (x) : \frac{4 sin ^{2} ( x ) + 4 π sin ( x ) + π ^{2} - 4 cos ^{2} ( x )}{4}

\frac{4 sin ^{2} ( x ) + 4 π sin ( x ) + π ^{2} - 4 cos ^{2} ( x )}{4} = 0

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

4 sin^{2} (x) + 4 π sin (x) + π^{2} - 4 cos^{2} (x) = 0

Rewrite using trig identities

- 4 + π^{2} + 8 sin^{2} (x) + 4 sin (x) π = 0

Solve by substitution

sin (x) = - \frac{π}{4} + i \frac{- 128 + 16 π ^{2}}{16}, sin (x) = - \frac{π}{4} - i \frac{- 128 + 16 π ^{2}}{16}

sin (x) = - \frac{π}{4} + i \frac{- 128 + 16 π ^{2}}{16} :

No Solution

sin (x) = - \frac{π}{4} - i \frac{- 128 + 16 π ^{2}}{16} :

No Solution

Combine all the solutions

No Solution

Verify solutions by plugging them into the original equation

No Solution for x \in R

5 cos (x) - 3 = 9 cos (x) - 5

3 sin^{2} (x) - 8 sin (x) + 4 = 0

3 tan (x) = tan (2 x)

cos (4 x) + sin (6 x) = 0

- 15.44 = - 2 sin (- 3 θ + 300)

What is the general solution for sin(x)+pi/2 =cos(x) ?
The general solution for sin(x)+pi/2 =cos(x) is No Solution for x\in\mathbb{R}