Question

Simplify the following, expressing as a single trigonometric function? cosec A + cot A? sin A + tan A?

I can easily simplify questions that are multiplication and division of Trig Identities but for some reason am missing something simple with the addition problems

Answer 1

`csc A + cot A = cot (A/2)`
`sin A + tan A = (4t)/(1 - t^4)`

Explanation:

a. `f(A) = 1/sin A + cos A/sin A = (1 + cos A)/sin A`
Note:
`1 + cos A = 2cos^2 (A/2)`
`sin A = 2sin (A/2).cos (A/2)`
`f(A) = (2cos^2 (A/2))/(2sin (A/2)cos (A/2)) = cos (A/2)/sin (A/2) = cot (A/2)`
2. f(A) = sin A + tan (A)
Call `tan (A/2) = t`
Use trig identities in terms of t -->
`sin A = (2t)/(1 + t^2)`
`tan A = (2t)/(1 - t^2)`
`f(t) = sin A + tan A = (2t)/(1 + t^2) + (2t)/(1 - t^2)`
`f(t) = (4t)/(1 - t^4)`