Simplify this compound fraction:
We start at the bottom.
cosAcscA=cosA1sinA=cosA⋅sinA\frac{cos A}{csc A} = \frac{cos A}{\frac{1}{sin A}} = cos A \cdot sin AcscAcosA=sinA1cosA=cosA⋅sinA.
cotAcosA⋅sinA=cosAsinA⋅1cosA⋅sinA=1sin2A\frac{cot A}{cosA \cdot sinA}=\frac{cos A}{sin A} \cdot \frac{1}{cos A \cdot sin A} = \frac{1}{{sin^2}A}cosA⋅sinAcotA=sinAcosA⋅cosA⋅sinA1=sin2A1.
secA1sin2A\frac{sec A}{\frac{1}{sin^2 A}}sin2A1secA = secA⋅sin2Asec A \cdot{sin^2 A}secA⋅sin2A = 1cosA⋅sin2A\frac{1}{cos A} \cdot {sin^2} AcosA1⋅sin2A = sin2AcosA\frac{{sin^2} A} {cos A}cosAsin2A.
tanAsin2AcosA\frac{tan A}{\frac{{sin^2} A} {cos A}}cosAsin2AtanA = sinAcosA⋅cosAsin2A\frac{sin A}{cos A} \cdot \frac{cos A}{{sin^2} A}cosAsinA⋅sin2AcosA = 1sinA\frac{1}{sin A}sinA1.
sinA1sinA=sin2A\frac{sin A}{\frac{1}{sin A}} = {sin^2 A}sinA1sinA=sin2A.
After all that!
