Question

If `tan θ = 4/3`, show that `sqrt((1 - sin θ)/(1 + sin θ)) = 1/3`.

Answer 1

Given: If `tan θ = 4/3`, show that `sqrt((1 - sin θ)/(1 + sin θ)) = 1/3`

Step-wise calculation:

1. Take a right triangle with opposite = 4 and adjacent = 3 so hypotenuse = `sqrt(4^2 + 3^2)` = 5.

2. Then sin θ = `"Opposite"/"Hypotenuse"`

= `4/5`

3. Substitute into the expression:

`sqrt((1 - sin θ)/(1 + sin θ))`

= `sqrt((1 - 4/5)/(1 + 4/5))`

= `sqrt((1/5)/(9/5))`

= `sqrt(1/9)`

= `1/3`

4. This uses the principal (positive) square root and the assumption that `sin θ = 4/5` i.e., θ is an acute angle. If θ were in a quadrant where `sin θ = -4/5`, the value would be 3 instead.

Hence, under the usual acute-angle assumption, `sqrt((1 - sin θ)/(1 + sin θ)) = 1/3`.