Recall that #cot(x)=cos(x)/sin(x)# and #sec(x)=1/cos(x)#.
#cot(A)=cos(A)/sin(A)#
#cot^2(A)=cos^2(A)/sin^2(A)#
#sec(A)=1/cos(A)#
#sec^2(A)=1/cos^2(A)#
If #sin(A)=3/5#, #sin^2(A)=(3/5)^2=9/25#
We have #sin^2(A)#; however, we still need #cos^2(A)# to determine the values of #cot^2(A)# and #sec^2(A).#
Recall the following identity:
#sin^2(x)+cos^2(x)=1#
So,
#sin^2(A)+cos^2(A)=1#
#cos^2(A)=1-sin^2(A)#
We know the value of #sin^2(A)#, so
#cos^2(A)=1-9/25=25/25-9/25=16/25#
#cot^2(A)=cos^2(A)/sin^2(A)=(16/25)/(9/25)=16/cancel25 * cancel25/9=16/9# (Because #(a/b)/(c/d)=a/b*d/c#)
#sec^2(A)=1/cos^2(A)=1/(16/25)=25/16# (Because #1/(a/b)=b/a#).
#cot^2(A)+sec^2(A)=16/9+25/16=32/16+25/16=57/16#
