# How do you find the value of theta for which cos(theta+pi/2) does not equal costheta+cos(pi/2)?

Jan 20, 2017

See explanation.

#### Explanation:

The inequality given is:

$\cos \left(x + \frac{\pi}{2}\right) \ne \cos x + \cos \left(\frac{\pi}{2}\right)$

This inequality can be transformed into:

$\cos \left(x + \frac{\pi}{2}\right) \ne \cos x$ because $\cos \left(\frac{\pi}{2}\right) = 0$

Now we can move $\cos x$ left:

$\cos \left(x + \frac{\pi}{2}\right) - \cos x \ne 0$

Now we use the formula

$\cos x - \cos y = - 2 \sin \left(\frac{x + y}{2}\right) \cdot \sin \left(\frac{x - y}{2}\right)$

$- 2 \sin \left(x + \frac{\pi}{4}\right) \sin \left(\frac{\pi}{4}\right) \ne 0$

$\sin \left(x + \frac{\pi}{4}\right) \ne 0$

This inequality is fulfilled for:

$x + \frac{\pi}{4} \ne k \cdot \pi$ for $k \in \mathbb{Z}$

$x \ne - \frac{\pi}{4} + k \cdot \pi$ for $k \in \mathbb{Z}$