Simplify and write the trigonometric expression in terms of sine and cosine:
sin x + (cot x)(cos x) = (1/f(x))
f(x)= ?
To simplify and write the trigonometric expression in terms of sine and cosine, let's start by rewriting the cotangent function in terms of sine and cosine using the identity: cot(x) = cos(x) / sin(x).
Substituting this in the original expression, we have:
sin(x) + (cos(x) / sin(x)) * cos(x) = 1 / f(x)
Now, let's simplify the expression:
sin(x) + (cos^2(x)) / sin(x) = 1 / f(x)
To combine the terms on the left side, we need a common denominator. The common denominator in this case is sin(x):
(sin(x) * sin(x)) / sin(x) + (cos^2(x)) / sin(x) = 1 / f(x)
Simplifying further, we have:
sin^2(x) + cos^2(x) / sin(x) = 1 / f(x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can substitute that in:
1 / sin(x) = 1 / f(x)
From this, we can conclude that f(x) = sin(x).
Therefore, the function f(x) equals sin(x).