Identities similar to $\arctan(x)+\arctan(1/x)=\pi/2$ The $\arctan(x)+\arctan(1/x)=\pi/2$ (for $x>0$) identity can be solved by taking the derivative of the left hand side, showing it is $0$, and then plugging in, say, $x=1$ to get its constant value $\pi/2$.
Are there any other (nontrivial) identities which can be solved similarly?  I am hoping for something a 1st semester Calculus student could solve... so not too difficult please!  

The proof for $x>0$ can be obtained as follows


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*let $\alpha=\arctan x \quad x\in\left(0,\frac{\pi}2\right)$
then
$$\tan\left(\frac{\pi}2-\alpha\right)=\frac1{\tan \alpha}=\frac1x \implies \frac{\pi}2-\alpha =\arctan \frac1x$$
Another similar identity is
$$\arcsin x + \arccos x=\frac{\pi}2 \quad \forall x\in[-1,1]$$
which can be proved by $\cos\left(\frac{\pi}2-\alpha\right)=\sin \alpha$.
Using derivatives we can prove some basic important inequalities as 


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*$\tan x\ge x\quad x\ge 0$

*$\sin x\le x\quad x\ge 0$

*$\sin x\ge x-x^3/6\quad x\ge 0$

*$\cos x \ge 1-\frac12 x^2$
and so on.
