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Help with a trig proof please?

#(cot u + csc u -1) / (cot u - csc u +1) = csc u + cot u#

Answer 1

Charles Autterson

Please see below.

#(cotu+cscu-1)/(cotu-cscu+1)#

= #(cotu+cscu-1)/(cotu-cscu+1)xx(cotu+cscu-1)/(cotu+cscu-1)#

= #(cotu+cscu-1)^2/(cot^2u-(cscu-1)^2)#

= #(cot^2u+csc^2u+1-2cscu-2cotu+2cscucotu)/(cot^2u-csc^2u-1+2cscu)#

= #(2csc^2u-2cscu-2cotu+2cscucotu)/(cot^2u-cot^2u-2+2cscu)#

= #(2cscu(cscu-1)+2cotu(cscu-1))/(2cscu-2)#

= #(2(cscu-1)(cscu+cotu))/(2(cscu-1))#

= #cscu+cotu#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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