Solve Pythagorean Identities Calc

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

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