SOLUTION: verify: tant+cost/1+sint=sec t tan2x+cot2x/sec2x=cscx csct-1/cott=cott/csct+1

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Question 714114: verify: tant+cost/1+sint=sec t
tan2x+cot2x/sec2x=cscx
csct-1/cott=cott/csct+1
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
I'll just do the first one:

tan(t) + cos%28t%29%2F%281%2Bsin%28t%29%29 = sec(t)

Rationalize the denominator of the second term by
multiplying it by %28conjugate_of_denominator%29%2F%28conjugate_of_denominator%29%22%22=%22%22%281-sin%28t%29%29%2F%281-sin%28t%29%29

tan(t) + cos%28t%29%2F%281%2Bsin%28t%29%29%22%22%2A%22%22%281-sin%28t%29%29%2F%281-sin%28t%29%29

tan(t) + %28cos%28t%29%281-sin%28t%29%29%29%2F%28%281%2Bsin%28t%29%29%281-sin%28t%29%29%29

tan(t) + %28cos%28t%29%281-sin%28t%29%29%29%2F%281-sin%5E2%28t%29%29

You know the identity sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1.  When it
is solved like this cos%5E2%28theta%29=1-sin%5E2%28theta%29 the right side
is the same form as the denominator and so we can write 1-sin%5E2%28t%29
as cos%5E2%28t%29

tan(t) + %28cos%28t%29%281-sin%28t%29%29%29%2Fcos%5E2%28t%29

And we cancel the cos(t) on top into the cosē(t) in the bottom

tan(t) + %28cross%28cos%28t%29%29%281-sin%28t%29%29%29%2Fcos%5Ecross%282%29%28t%29  

tan(t) + %281-sin%28t%29%29%2Fcos%28t%29

Now we substitute sin%28t%29%2Fcos%28t%29 for tan%28t%29:

sin%28t%29%2Fcos%28t%29%22%22%2B%22%22%281-sin%28t%29%29%2Fcos%28t%29

Combine the fractions (they already have a common denominator):

%28sin%28t%29%2B1-sin%28t%29%29%2Fcos%28t%29

%28cross%28sin%28t%29%29%2B1-cross%28sin%28t%29%29%29%2Fcos%28t%29

1%2Fcos%28t%29

sec%28t%29

Edwin