Prove the Pythagorean identities in trigonometry.

The Pythagorean identities in trigonometry are proven using the unit circle and basic algebraic manipulation.

The first Pythagorean identity is sin²θ + cos²θ = 1. To prove this, we start with the unit circle, where the point (cosθ, sinθ) lies on the circle with radius 1. We can then use the Pythagorean theorem to find the length of the hypotenuse of the right triangle formed by the x-axis, y-axis, and the point (cosθ, sinθ). This gives us:

cos²θ + sin²θ = 1

We can then rearrange this equation to get the first Pythagorean identity:

sin²θ + cos²θ = 1

The second Pythagorean identity is tan²θ + 1 = sec²θ. To prove this, we start with the definition of tangent and secant:

tanθ = sinθ/cosθ
secθ = 1/cosθ

We can then substitute these definitions into the left-hand side of the equation:

tan²θ + 1 = (sin²θ/cos²θ) + 1
= (sin²θ + cos²θ)/cos²θ
= 1/cos²θ

We can then substitute the definition of secant into the right-hand side of the equation:

sec²θ = (1/cosθ)²
= 1/cos²θ

Therefore, we have proven the second Pythagorean identity:

tan²θ + 1 = sec²θ