Question

need help verifing this...
cosx + senx + ctgx = cscx

Answer 1

Question 32904

$$\cos x + \sin x + \cot x = \csc x$$

This identity is not equal for all $x$ values.

One might solve it, by first rewriting it as $\cos x + \sin x + \frac{\cos x}{\sin x} - \frac{1}{\sin x} = 0$, or

$\cos x + \sin x = \frac{1 - \cos x}{\sin x}$ . Multiplying both sides by $\sin x \neq 0$ , obtain

$\cos x \sin x + \sin^2 x = 1 - \cos x$ , or $\cos x \sin x - \cos^2 x = -\cos x$ , from which $\sin x - \cos x = -1$ .

Formally, this equation has two roots $x = 0$ and $x = -\frac{\pi}{2}$ .

But substituting it into $\cos x + \sin x + \cot x = \csc x$ , check that only $x = -\frac{\pi}{2}$ is the solution.