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 the trigonometric expression and write it in terms of sine and cosine, we will use the trigonometric identities. Let's start by writing down the given expression:
sin x + (cot x)(cos x) = 1/f(x)
First, let's express the cotangent in terms of sine and cosine:
cot x = cos x / sin x
Now let's substitute this value into the expression:
sin x + (cos x / sin x)(cos x) = 1/f(x)
To simplify further, we need to eliminate the fraction denominator, sin x. We can do this by multiplying both sides of the equation by sin x:
sin x * sin x + cos x * cos x = sin x * (1/f(x))
Using the Pythagorean identity sin^2 x + cos^2 x = 1, we can simplify the left side of the equation:
1 = sin x * (1/f(x))
Now we can isolate f(x) by dividing both sides of the equation by sin x:
1/f(x) = 1 / sin x
Therefore, f(x) = sin x.
So, when simplified and written in terms of sine and cosine, the trigonometric expression is f(x) = sin x.