Prove that (1+sec(-x))/(sin(-x)+tan(-x)) equals -csc(x)

(1+sec(-x))/(sin(-x)+tan(-x))
= (1 + 1/cos(-x))/(sin(-x) + sin(-x)/cos(-x)), using definitions of secant and tangent
= (1 + 1/cos(x))/(-sin(x) - sin(x)/cos(x)), since sine is odd and cosine is even
= (cos(x) + 1)/(-sin(x)cos(x) - sin(x)), from multiplying top and bottom by cos(x)
= (cos(x) + 1)/(-sin(x)(cos(x) + 1)), from factoring out -sin(x) in the denominator
= -1/sin(x), from dividing top and bottom by (cos(x) + 1)
= -csc(x), using the definition of cosecant

Lord bless you today!

A simpler way:
(1+sec(-x))/(sin(-x)+tan(-x)…
= (1+sec(-x))/[sin(-x) (1+sec(-x))]
=1/sin(-x)
= -csc(x)

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