Question

If `sec θ = 5/4`, evaluate `(sin θ - 2 cos θ)/(tan θ - cot θ)`.

Answer 1

Given: `sec θ = 5/4`. So `cos θ = 4/5`.

Step-wise calculation:

1. From `sec θ = 5/4`, `cos θ = 4/5`.

If θ is acute (standard assumption), `sin θ = 3/5`.

2. Numerator:

`sin θ - 2 cos θ = 3/5 - 2 xx (4/5)`

= `3/5 - 8/5`

= `-5/5`

= –1

3. Denominator:

`tan θ - cot θ = (sin/cos) - (cos/sin)`

= `3/4 - 4/3`

= `(9 - 16)/12`

= `(-7)/12`

4. Fraction:

`(sin θ - 2 cos θ)/(tan θ - cot θ)`

= `(-1)/(-7/12)`

= `12/7`

If θ were not acute so that `sin θ = −3/5`, the same steps give the value `-132/35`; therefore the expression depends on the sign of sin θ. For the typical acute-angle interpretation the value is `12/7`.

`12/7` assuming θ is acute.