Question

Hi
Trigonometry identity dirivation

Answer 1

Therefore, on dividing both numerator and denominator by r,

tan θ = y/r

x/r = sin θ

cos θ .

cot θ = 1

tan θ = cos θ

sin θ .

Those are the two identities.

Proof of the Pythagorean identities

To prove:

a) sin2θ + cos2θ = 1

b) 1 + tan2θ = sec2θ

c) 1 + cot2θ = csc 2θ

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.

That is, according to the definitions,

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 three Pythagorean identities are thus equivalent to one another.