Question

If `sec θ = 5/4`, verify that `(tan θ)/(1 + tan^2θ) = (sin θ)/(sec θ)`.

Answer 1

Given: `sec θ = 5/4`

Step-wise calculation:

1. `cos θ = 1/(sec θ)` 

= `4/5`

2. `sin θ = sqrt(1 - cos^2θ)`

= `sqrt(1 - (4/5)^2)`

= `sqrt(1 - 16/25)`

= `sqrt(9/25)`

= `3/5` (θ acute).

3. `tan θ = (sin θ)/(cos θ)`

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

= `3/4`

4. Left side:

`(tan θ)/(1 + tan^2θ) = (3/4)/(1 + 9/16)` 

= `(3/4)/(25/16)`

= `(3/4) xx (16/25)`

= `12/25`

Alternatively use identity 1 + tan²θ = sec²θ.

So, `tan/(1 + tan^2) = tan/(sec^2)`

= `(sin/cos) · cos^2`

= sin · cos

5. Right side:

`(sin θ)/(sec θ) = (3/5) / (5/4)` 

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

= `12/25`

`(tan θ)/(1 + tan^2 θ) = (sin θ)/(sec θ) = 12/25`, so the equality is verified.