Simplify and write the trigonometric expression in terms of sine and cosine: cot(-x)cos(-x)+sin(-x)=-1/f(x)=f(x)=____________
I've tried everything, I think that the "-1/f(x)" part is screwing up my answer but I'm not sure what it's asking me to do with that?
thank you, the answer is just:
=sin(x)
To simplify and write the trigonometric expression in terms of sine and cosine, we'll work step-by-step:
1. Start with the given expression:
cot(-x)cos(-x) + sin(-x) = -1/f(x)
2. Recognize that cot(-x) and cos(-x) are equivalent to their positive counterparts:
cot(-x) = cot(x) and cos(-x) = cos(x)
3. Express the given expression using the positive angles:
cot(x)cos(x) + sin(-x) = -1/f(x)
4. Use the Pythagorean identity to rewrite the cotangent term:
cot(x)cos(x) = (cos(x)/sin(x))cos(x) = cos^2(x)/sin(x) = 1/sin(x) - sin(x)
5. Substitute the simplified cotangent expression back into the original expression:
1/sin(x) - sin(x) + sin(-x) = -1/f(x)
6. Simplify the expression by using the identity sin(-x) = -sin(x):
1/sin(x) - sin(x) - sin(x) = -1/f(x)
7. Combine like terms:
1/sin(x) - 2sin(x) = -1/f(x)
8. Multiply both sides of the equation by sin(x) to get rid of the denominator:
1 - 2sin^2(x) = -sin(x)f(x)
9. Rearrange the equation to isolate the term -sin(x)f(x) on one side:
-2sin^2(x) - sin(x)f(x) + 1 = 0
10. Finally, substitute f(x) as the answer:
f(x) = -2sin^2(x) - sin(x) + 1
No, it isn't
the simplification of the expression is
-1/sinx, as you can see from my steps
however,
if -1/f(x) = -1/sinx
then f(x) = sinx
I will "simplify and write the trigonometric expression in terms of sine and cosine"
the part "=-1/f(x)=f(x)=____________ "
makes little sense to me
cot(-x)cos(-x)+sin(-x)
= (cos(-x))/sin(-x))(cos(-x) + sin(-x)
= (cosx/-sinx)(cosx) - sinx
= - cos^2 x/sinx - sin^2 x/sinx
= -(cos^2 x + sin^2 x)/sinx
= -1/sinx