Use a right triangle to write cot(sin−1(√x2−25/x)) as an algebraic expression. assume x is positive

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Answer 1

cot(θ) = cos(θ)/sin(θ)

The denominator is sin(sin^-1(2√x - 25/x) = 2√x - 25/x, because a function of its inverse yields the input.

For the numerator, we need to find cos(sin^-1(2√x - 25/x)).

We need to find cos(θ), where sin(θ) = 2√x - 25/x.

Use the trig identity sin²(θ) + cos²(θ) = 1: cos²(θ) = 1 - sin²(θ)

cos²(θ) = 1 - (2√x - 25/x)² = 1 - 4x + (100√x)/x + 625/x²

coa(θ) = √(1 - 4x + (100√x)/x + 625/x²)

Then cot(θ) = √(1 - 4x + (100√x)/x + 625/x²)/(2√x - 25/x),