Answer:
Step-by-step explanation
There are many ways to prove this identity. But, I will use the simplest one. If you have a right triangle, the length of the horizontal side is x, the length of the vertical side is y, and the hypotenuse is r. So, the relation between x and r can be written in cosine form:
[tex]\cos\theta=\frac{x}{r}[/tex]
The relation between y and r can be written in sine form:
[tex]\sin\theta=\frac{y}{r}[/tex]
Pythagoras told you that for the right triangle [tex]x^{2}+y^{2}=r^{2}[/tex] should be satisfied. Substitute x and y from the cosine and sine form:
[tex](r\cos\theta)^{2}+(r\sin\theta)^{2}=r^{2}[/tex]
[tex]r^{2}\cos^{2}\theta+r^{2}\sin^{2}\theta =r^{2}[/tex]
the square of r will cancel out, and you can get your identity:
[tex]\cos^{2}\theta+\sin^{2}\theta=1[/tex]
