Prove that cos(sin inverse x) = square root ( 1-x^2)
To prove that cos(sin^(-1)(x)) = sqrt(1 - x^2), we can use the trigonometric identity sin^(-1)(x) = arcsin(x).
Step 1: Start with the equation cos(sin^(-1)(x)) = sqrt(1 - x^2).
Step 2: Rewrite sin^(-1)(x) as arcsin(x), so the equation becomes cos(arcsin(x)) = sqrt(1 - x^2).
Step 3: Recall the definition of arcsin(x): arcsin(x) is the angle whose sine is x, within the range of -π/2 to π/2.
Step 4: Consider a right triangle with an angle θ and sides adjacent (a), opposite (b), and hypotenuse (c).
Step 5: Let's assume that sin(θ) = x, which means that opposite side b is x and the hypotenuse c is 1.
Step 6: Using the Pythagorean theorem, a^2 + x^2 = 1^2, which simplifies to a^2 = 1 - x^2.
Step 7: From the definition of cosine (cos(θ) = a/c), we have cos(arcsin(x)) = a/1 = a.
Step 8: Substitute the value of a from step 6 into cos(arcsin(x)), yielding cos(arcsin(x)) = sqrt(1 - x^2).
Step 9: The equation cos(sin^(-1)(x)) = sqrt(1 - x^2) has been proven.
Therefore, cos(sin^(-1)(x)) is equal to sqrt(1 - x^2).