SOLUTION: Simplify the trigonometric expression. Once you have simplified the expression completely, change all trigonometric functions in the simplified expression to sines and cosines. Fin

Algebra ->  Trigonometry-basics -> SOLUTION: Simplify the trigonometric expression. Once you have simplified the expression completely, change all trigonometric functions in the simplified expression to sines and cosines. Fin      Log On


   

Question 924637: Simplify the trigonometric expression. Once you have simplified the expression completely, change all trigonometric functions in the simplified expression to sines and cosines. Finally, let s = sin θ and c = cos θ and enter your answer below. For example, if the expression simplifies to tan θ, you would enter s/c in the answer blank.
csc(θ) + cos(-θ)cot(-θ) =
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Use the definitions and go from that.

1%2Fsin%28t%29%2Bcos%28-t%29%28cos%28-t%29%2Fsin%28-t%29%29

Deal with the negative signs in the functions first.
Refer to the Unit Circle for visual help.

1%2Fsin%28t%29%2Bcos%28t%29%28cos%28t%29%2F%28-sin%28t%29%29%29

1%2Fsin%28t%29-cos%5E2%28t%29%2Fsin%28t%29, notice common denominators.

%281-cos%5E2%28t%29%29%2Fsin%28t%29, justification here for the basic Pythagorean Thm. Identity

sin%5E2%28t%29%2Fsin%28t%29

highlight%28sin%28t%29%29.


The other described substitutions are not really needed; but they COULD be used when you finally have all function inputs as the positive t, which I chose to use instead of " t h e t a ". (We would need to spell it with all five letters to make the rendering work.)