You are given that #cos(A)=5/13#. That is, for a triangle, the adjacent, "adj", angle is #5# and the hypotenuse, "hyp", is #13#.
Using the Pythagorean Theorem, you can get the opposite side
#"adjacent"^2+"opposite"^2="hypotenuse"^2#
#(5)^2+"opposite"^2=(13)^2#
#25+"opp"^2=169#
#"opp"^2=144#
#"opp"=12#
Using the right-triangle rules of trigonometric functions we have:
#sin(A)=("opp")/("hyp")=12/13#
#cos(A)=("adj")/("hyp")=5/13#
#tan(A)=("opp")/("adj")=12/5#
#csc(A)=("hyp")/("opp")=13/12#
#sec(A)=("hyp")/("adj")=13/5#
#cot(A)=("adj")/("opp")=5/12#
Plug these values into the first question:
#sin(A)-cot(A)/(2tan(A))=12/13-(5/12)/(2xx12/5)=12/13-5/12xx5/24#
#=12/13-25/288=3131/3744~~0.8363#
Finally, plug our values into the second question:
#cot(A)+1/(cos(A))=5/12+1/(5/13)=5/12+13/5=181/60~~3.017#
