Question 1156990: Write the trigonometric expression in terms of sine and cosine, and then simplify.
sin(θ) − csc(θ)
_______________
cos(θ)
Found 3 solutions by Theo, Boreal, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = theta.
this is not necessary, but easier to type.
problem becomes:
(sin(x) - csc(x) / cos(x)
csc(x) = 1/sin(x), therefore the expression becomes:
(sin(x) - 1/sin(x)) / cos(x)
multiply numerator and denominator by sin(x) to get:
(sin^2(x) - 1) / (sin(x)cos(x))
sin^2(x) - 1 = -(1 - sin^2(x) = -cos^2(x), therefore the expression becomes:
-cos^2(x) / (sin(x)cos(x))
divide numerator and denominator by cos(x) to get:
-cos(x) / sin(x)
since this is equivalent to -cot(x), then you have:
(sin(x) - csc(x) / cos(x) = -cot(x)
that's your solution, as far as i can see.
confirm by assigning any angle that pops into your head.
i'm working in degrees, so make sure your calculator is working in degrees if you want to try this yourself.
i chose 3 angles at random.
-cot(x) is confirmed to be equivalent to (sin(x) - csc(x)) / cos(x) for all 3 of them.
for example, if i chose 257936 degrees:
(sin(x) - csc(x) / cos(x) = 14.30066626
-cot(x) = 14.30066626
sine they're the same, the two expressions are equivalent.
note that csc(x) = 1/sin(x) and cot(x) = 1/tan(x).
if your calculator can't handle csc(x) or cot(x) directly, you will need to use those equivalents.
Answer by Boreal(15235)  (Show Source):
You can put this solution on YOUR website! csc x=(1/sin x)
so numerator is sin x-(1/sin x) or (sin^x-1)/sin x
but cos^2x=1-sin^2x
so the numerator is -cos^2x/sin x
That is divided by cos x
and the answer is -cos x/ sin x
or -ctn x.
Answer by MathTherapy(10551)  (Show Source):
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