Question

Solve the equation sin²x + cosx=2​

Answer 1

Recall the Pythagorean identity,

`\sin^2x+\cos^2x=1`

So the equation is the same as

`(1-\cos^2x)+\cos x=2\implies\cos^2x-\cos x+1=0`

Complete the square:

`\cos^2x-\cos x+\frac14+\frac34=0`

`\left(\cos x-\frac12\right)^2+\frac34=0`

`\implies\left(\cos x-\frac12\right)^2=-\frac34`

But whenever you square a real number, you get a positive number, so there are no real solutions to this equation.