MCQ: comparing two quantities; $\tan(1)$ and $\frac{\pi}{2}$

Question

Without using a calculator, given that $\pi \approx 3.14159$, compare:

FIRST VALUE: $\tan(1)$ [the angle is in radians].

SECOND VALUE: $\frac{\pi}{2}$

Options:

(A) FIRST VALUE is greater than the SECOND VALUE

(B) FIRST VALUE is less than the SECOND VALUE

(C) FIRST VALUE is equal to the SECOND VALUE

(D) The two quantities cannot be compared with the given information.

My thinking:

Expanding

$\tan(x)=x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+\dots$

Putting $x=1$, we get $\tan(1)=1+\frac{1}{3}+\frac{2}{15}+\frac{17}{315}+\dots$

So $\tan(1)=1.52063+$ other positive terms

But $\frac{\pi}{2} \approx 1.57\dots$

However, we do not know if that "+ other positive terms" will make it greater than $1.57\dots$ or no.

How can we be sure?

Is there a better way?

Any help will be appreciated. THANKS!

Answer 1

Hint: try to apply $\arctan$ to both values...