Simplify and write the trigonometric expression in terms of sine and cosine:
sin x + (cot x)(cos x) = (1/f(x))
f(x)= ?
cot = cos/sin
sin x + cos^2 x /sin x = 1/y
sin^2 x + cos^2 x = sin x / y
but sin^2 + cos^2 = 1
so
y = sin x
To simplify and write the trigonometric expression in terms of sine and cosine, we need to eliminate the cotangent term.
Recall the identities:
cot x = cos x / sin x
and
1/f(x) = cot x
By substituting cot x using the identity, we get:
sin x + (cos x / sin x) * cos x = 1 / f(x)
To simplify further, we need to clear the fraction by multiplying both sides of the equation by sin x:
sin x * sin x + cos x * cos x = sin x / f(x)
Using the Pythagorean identity:
sin² x + cos² x = 1
We can rewrite the expression as:
1 = sin x / f(x)
Now, to find f(x), we need to isolate it. We can do this by cross-multiplying:
sin x = f(x)
Therefore, f(x) = sin x.