Compare the following numbers...

Question

How can I compare this two numbers: $\tan \frac 7{10}$ and $1$?

Hope your explanation will be as explicit as possible! Thank you very much!

Answer 1

The tangent function is an increasing function on $(-\pi/2, \pi/2)$. Also, $\frac{7}{10} < \frac{3}{4} < \frac{\pi}{4}$, so $\tan(7/10) < \tan(\pi/4)$.

Answer 2

Since on the interval $\left(\dfrac{\pi}{2},-\dfrac{\pi}{2}\right)$ the function $\tan(x)$ has positive slope, if $a>b,\tan(a)>\tan(b)\forall a,b\in\left(\dfrac{\pi}{2},-\dfrac{\pi}{2}\right)$. Therefore, since $\dfrac{7}{10}<\dfrac{\pi}{4}$ and $\tan\left(\dfrac{\pi}{4}\right)=1$, we may say that $$\tan\left(\dfrac{7}{10}\right)<\tan\left(\dfrac{\pi}{4}\right)$$