Answer :
To prove that x^2 + y^2 = 1 when cos^-1(x) * cos^-1(y) = π/2, we can use the properties of trigonometric functions.
Given:
cos^-1(x) * cos^-1(y) = π/2
Step 1: Understand the relationship between cos^-1(x) and x.
cos^-1(x) is the inverse cosine function, which means that it is the angle whose cosine is x.
Therefore, cos(cos^-1(x)) = x
Step 2: Apply the given condition.
cos^-1(x) * cos^-1(y) = π/2
Substituting the relationship from Step 1:
cos(cos^-1(x)) * cos(cos^-1(y)) = cos(π/2)
x * y = 0
Step 3: Solve for x^2 + y^2 = 1.
Since x * y = 0, we can conclude that either x = 0 or y = 0.
If x = 0, then y^2 = 1, and x^2 + y^2 = 1.
If y = 0, then x^2 = 1, and x^2 + y^2 = 1.
Therefore, we have proven that when cos^-1(x) * cos^-1(y) = π/2, then x^2 + y^2 = 1.
Given:
cos^-1(x) * cos^-1(y) = π/2
Step 1: Understand the relationship between cos^-1(x) and x.
cos^-1(x) is the inverse cosine function, which means that it is the angle whose cosine is x.
Therefore, cos(cos^-1(x)) = x
Step 2: Apply the given condition.
cos^-1(x) * cos^-1(y) = π/2
Substituting the relationship from Step 1:
cos(cos^-1(x)) * cos(cos^-1(y)) = cos(π/2)
x * y = 0
Step 3: Solve for x^2 + y^2 = 1.
Since x * y = 0, we can conclude that either x = 0 or y = 0.
If x = 0, then y^2 = 1, and x^2 + y^2 = 1.
If y = 0, then x^2 = 1, and x^2 + y^2 = 1.
Therefore, we have proven that when cos^-1(x) * cos^-1(y) = π/2, then x^2 + y^2 = 1.
