Question 41 of 42intermediate🔧 ApplyMCQ1 mark

sin(tan⁻¹x), |x| < 1 is equal to

Exercise: Miscellaneous Exercise on Chapter 2 | Q: 13 | (Page 14)
For More Understanding

Explanation

To find sin(tan⁻¹x), we use the substitution method. Let tan⁻¹x = θ, which means tan θ = x. We need to find sin θ. Using a right triangle approach: if tan θ = x/1, then opposite = x, adjacent = 1, and hypotenuse = x2+1\sqrt{x² + 1}. Therefore, sin θ = opposite/hypotenuse = x/1+x2\sqrt{1 + x²}. This matches option (D).

Solution Steps

  1. Step 1: Let tan⁻¹x = θ, so tan θ = x

  2. Step 2: Construct a right triangle where tan θ = x/1, meaning opposite = x and adjacent = 1

  3. Step 3: Calculate hypotenuse = x2+1\sqrt{x² + 1}

  4. Step 4: Find sin θ = opposite/hypotenuse = x/1+x2\sqrt{1 + x²}

  5. Step 5: Therefore, sin(tan⁻¹x) = x/1+x2\sqrt{1 + x²}, which is option (D)

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