Question

If `tan A = sqrt(3)`, where A is an acute angle, then find the value of `(sin^2 A)/(1 + cos^2 A)`.

Answer 1

Given: `tan A = sqrt(3)`, A is acute.

Step-wise calculation:

1. `tan A = sqrt(3)`

⇒ Take a right triangle with opposite = `sqrt(3)`, adjacent = 1.

2. Hypotenuse = `sqrt((sqrt3)^2 + 1^2)`

= `sqrt(4)` 

= 2

3. `sin A = "Opposite"/"Hypotenuse"`

= `(sqrt(3))/2`

So `sin^2 A = 3/4`.

4. `cos A = "Adjacent"/"Hypotenuse" = 1/2`

So `cos^2 A = 1/4`.

5. `(sin^2 A)/(1 + cos^2 A) = (3/4)/(1 + 1/4)`

= `(3/4)/(5/4)`

= `3/5`