how do i prove that $\sin(\pi/4)=\cos(\pi/4)$?

Question

It's weird that i have not defined the tangent function yet.

how do i prove that $\sin(\pi/4)=\cos(\pi/4)$?

I have prove that $\tan:(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ is a strictly increasing continuous bijection. (Not yet proved that it's a homeomorphism; i think i can show that arctan is concave on $(0,\infty)$)

Anyhow, i have no idea how do prove that $\tan(\pi/4)=1$.. Please help

Answer 1

Hint: Trigonometry is all about triangles. In this case, the triangle is an isosceles right angled triangle. Now, think about sine and cosine in respect of sides of triangle. I think you have found your necessary proof.

Proof 2: $\sin \dfrac{\pi}{4}=\cos\bigg(\dfrac{\pi}{2}-\dfrac{\pi}{4}\bigg)=\cos \dfrac{\pi}{4}$

Hint 3: Try to see the symmetry of the graphs of sine and cosine function in the interval $[0,\frac{\pi}{2}]$. By observing the symmetry you can find that the two graphs cut at the mid-point of the given interval so they take the same value at that specific point.

Answer 2

How about simply showing that $\sin(\pi/4)=\frac{\sqrt 2}{2}=\cos(\pi/4)$?