sin2θ+cos2θ=1
1+tan2θ=sec2θ
1+cot2θ=csc 2θ
Pythagorean identities
a) | sin2θ + cos2θ | = | 1. |
b) | 1 + tan2θ | = | sec2θ |
c) | 1 + cot2θ | = | csc 2θ |
Proof of the Pythagorean identities
Proof 1. According to the Pythagorean theorem,
x2 + y2 = r2. . . . . . . . . . . . . . . .(1)
Therefore, on dividing both sides by r2,
x2 r2 |
+ | y2 r2 |
= | r2 r2 |
= 1. |
cos2θ + sin2θ = 1. . . . . . . . . . . . . .(2)
Apart from the order of the terms, this is the first Pythagorean identity, a)
To derive b), divide line (1) by x2; to derive c), divide by y2.
Or, we can derive both b) and c) from a) by dividing it first by cos2θ and then by sin2θ. On dividing line 2) by cos2θ, we have
That is,
1 + tan2θ = sec2θ.
And if we divide a) by sin2θ, we have
That is,
1 + cot2θ = csc2θ.
The 3 Pythagorean identities are thus equivalent to one another.
Proof 2.
sin2θ + cos2θ | = | y2 r2 |
+ | x2 r2 |
= | y2 + x2 r2 |
= | r2 r2 |
|
According to the Pythagorean theorem, = 1 |