1. This one is tricky....you have to multiply the top and bottom by the following term: 1-cscx
When you do this, you get the following: 1-csc^2 x/[cotx*(1-cscx)]
We know that 1+cot^2 x = csc^2 x....That means 1-csc^2 x is equal to -cot^2 x.
So we have -cot^2 x/[cotx*(1-cscx)]...one cotx term cancels and you get -cotx/(1-cscx)...removing the negative sign from the top and bottom yields cotx/(cscx-1)
We know that cotx = cosx/sinx....so multiply the numerator and denominator by sinx...this yields:
cosx/(cscx*sinx -sinx)....since sinx = 1/cscx...that leaves us with cosx/(1-sinx)
2.
Find the common denominator...which is (1+sinx)cosx
That yields:
cos^2 x / [(1+sinx)cosx] for the first term and (1+2sinx + sin^2 x)/[(1+sinx)cosx] for the second
The numerator then becomes (1+2sinx + sin^2 x) + cos^2 x....since sin^2 x + cos^2 x = 1....this reduces to 1+2sinx+1 which further reduces to 2(1+sinx)
Thus, we have 2(1+sinx)/[(1+sinx)cosx]....the 1+sinx terms cancel out which yields 2/cosx which is also written as 2secx
This post was edited by thundercock on Apr 1 2013 12:21am