Precalculus/Trig

I can't seem to prove these trig identities and would really appreciate help:

1. cosx + 1/sin^3x = cscx/1 - cosx

I changed the 1: cosx/sin^3x + sin^3x/sin^3x = cscx/1-cosx
Simplified: cosx + sin^3x/sin^3x = cscx/1-cosx
I don't know where to go from there.

2. (sinx + cosx)(tanx + cotx) = secx + cscx

I made this into (sinx + cosx)(sinx/cosx + cosx/sinx) = secx + cscx
FOILed: sin^2x/cosx + sinxcosx/sinx + sinxcosx/cosx + cos^2x/sinx
Did I go wrong somewhere? What do I do next?

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  1. (cos+1)/sin^3 = csc/(1-cos)
    multiply by (1-cos)
    (1+cos)(1-cos)/sin^3 = csc = 1/sin
    (1-cos^2)/sin^2 = 1
    sin^2/sin^2 = 1
    1 = 1

    (sin+cos)(tan+cot) = sec+csc
    sin*tan + cos*tan + sin*cot + cos*cot = sec+csc
    sin^2/cos + sin + cos + cos^2/sin = sec+csc
    (sin^2+cos)/cos + (sin^2+cos^2)/sin = sec+csc
    1/cos + 1/sin = sec+csc
    sec+csc = sec+csc

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