Chapter 8 · Question 5

Prove the identity $\sin^2 A+\cos^2 A=1$ .

Answer

In a right triangle,

\sin A=\frac{p}{h}

and

\cos A=\frac{b}{h}

. Then

\sin^2A+\cos^2A=\frac{p^2+b^2}{h^2}=\frac{h^2}{h^2}=1

by Pythagoras theorem.

Square sine and cosine. The numerator becomes hypotenuse squared, so the result is 1.

Let perpendicular

=p

, base

=b

, hypotenuse

=h

. Then

\sin A=\frac ph

and

\cos A=\frac bh

. Hence

\sin^2A+\cos^2A=\frac{p^2}{h^2}+\frac{b^2}{h^2}=\frac{p^2+b^2}{h^2}

. Since

p^2+b^2=h^2

, the value is

1

.