Question

Express in terms of x and y:sin(cos⁻¹x+tan⁻¹y)

Answer 1

To express sin(cos⁻¹x+tan⁻¹y) in terms of x and y, use the trigonometric identities sin(cos⁻¹x) = √(1-x²) and tan⁻¹y = arctan(y). Apply the sum of angle formula to find the value of sin(θ).

Step-by-step explanation:

To express sin(cos⁻¹x+tan⁻¹y) in terms of x and y, we can use the trigonometric identities sin(cos⁻¹x) = √(1-x²) and tan⁻¹y = arctan(y). So we have sin(cos⁻¹x+tan⁻¹y) = sin(cos⁻¹x+arctan(y)).

Let's call the angle cos⁻¹x + arctan(y) as θ. Using the sum of angle formula, we have sin(θ) = sin(cos⁻¹x)cos(arctan(y)) + cos(cos⁻¹x)sin(arctan(y)).

Since cos⁻¹x and arctan(y) are angles where x and y are the adjacent and opposite sides respectively of a right triangle, we can express them in terms of x and y using trigonometric ratios.