Sine and cosine problem?

Question

I need to prove how AB = 1
when Asin0+cos0=1 and Bsin0-cos0=1

0 being phaeta. (oO" don't know how to make it any other way.)

Answer 1

Rearranging the two eq and I am using x instead of O

A = (1-cosx)/sinx and B=(1+cosx)/sinx

AB = (1-cosx)(1+cosx)/sin^2 x
= (1-cos^2x)/sin^2x
= sin^2x / sin^2 x
= 1

Answer 2

Let
Asinθ + cosθ = 1
and
Bsinθ - cosθ = 1

Add the equations to eliminate cosθ and solve for sinθ.
Asinθ + cosθ = 1
Bsinθ - cosθ = 1
(A + B)sinθ = 2
sinθ = 2/(A + B)

Substitute that solution into either of the given relations and solve for cosθ.
A[2/(A + B)] + cosθ = 1
2A/(A + B) + cosθ = 1
cosθ = 1 - 2A/(A + B)

From the identity 1 = cos²θ + sin²θ
1 = [1 - 2A/(A + B)]² + [2/(A + B)]²
1 = [(A + B)/(A + B) - 2A/(A + B)]² + [2/(A + B)]²
1 = [(A + B) - 2A]²/(A + B)² + 2²/(A + B)²
(A + B)² = (B – A)² + 4
A² + 2AB + B² = B² - 2AB + A² + 4
4AB = 4
AB = 1

Answer 3

First solve each equation for A and B
A = (1 - cosO) / sinO and B = (1 + cosO)/ sinO

Multiplying AB = (1-cosO)(1+cosO)/sin^2(O)
= (1-cos^2(O)) / sin^2(O) using a pyghag. identity, now get
= sin^2(O)/sin^2(O)
= 1