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There are many ways to do it, sometimes its easier only to work on one side of the equation, sometimes on the other, sometimes on both at the same time. Here is a exemple only working on the left side of the equation:
With csc^2=1+cot^2 and sin^2+cos^2=1:
1/(1+cos) + (csc^2 -1)/cos = **(Note: csc^2=1+cot^2)
1/(1+cos) + cot^2/cos =
1/(1+cos) + cos/sin^2 =
sin^2/(1+cos)*(sin^2) + (1+cos)*cos/(1+cos)*(sin^2) =
sin^2 +cos^2 +cos / sin^2*(1+cos) =
(1+cos) /sin^2*(1+cos) = **(Note: sin^2+cos^2=1)
Hope it will help. Good luck.