calculus - Limits with trigonometric functions without using L'Hospital Rule.

I want to find the limits $$\lim_{x\to \pi/2} \frac{\cos x}{x-\pi/2} $$
and
$$\lim_{x\to\pi/4} \frac{\cot x - 1}{x-\pi/4} $$

and
$$\lim_{h\to0} \frac{\sin^2(\pi/4+h)-\frac{1}{2}}{h}$$
without L'Hospital's Rule.

I know the fundamental limits $$\lim_{x\to 0} \frac{\sin x}{x} = 1,\quad \lim_{x\to 0} \frac{\cos x - 1}{x} = 0 $$

Progress

Using $\cos x=\sin\bigg(\dfrac\pi2-x\bigg)$ I got $-1$ for the first limit.