Question

Cot2A+tanA=?

Answer 1

`csc2A`

Explanation:

`cot2A+tanA`

`=(cos2A)/(sin2A)+sinA/cosA`

`=(cos2AcosA+sin2AsinA)/(sin2AcosA)`

`=cos(2A-A) /(sin2AcosA)`

`=cosA/(sin2AcosA)`

`=1/(sin2A)`

`=csc2A`

Formulae:

• `cosCcosD+sinCsinD=cos(C-D)`
• `cos2AcosA+sin2AsinA=cos(2A-A)`

Answer 2

= csc 2A

Explanation:

Call tan A = t , and apply the trig identity
`tan 2A = (2tan A)/(1 - tan^2 A)`
The expression becomes:
`f (A) = cot 2A + tan A = (1 - t^2)/(2t) + t = ((1 - t^2) + 2t^2)/(2t)`
`f(A) = (1 + t^2)/(2t) = (1 +tan^2 A)/(2tan A)`
Using trig identity, replace `(1 + tan^2 A)` by `(sec^2 A)`, we get:
`f(A) = (sec^2 A)((2sin A)/(cos A)) = (cos A)/(2sin A.cos^2 A)`
Note that: `sin 2A = 2sin A.cos A` -->
`f(A) = 1/(sin 2A) = csc 2A`