Final answer:
To simplify the complex fraction (1)/(tanx+cotx), first express tanx and cotx in terms of sine and cosine functions. Find a common denominator and simplify the expression to sinx*cosx.
Step-by-step explanation:
To simplify the complex fraction (1)/(tanx+cotx), we can start by expressing tanx and cotx in terms of sine and cosine functions. tanx is equal to sinx/cosx, and cotx is equal to cosx/sinx. Substituting these values into the complex fraction, we get:
(1)/((sinx/cosx)+(cosx/sinx)).
Next, we can find a common denominator for the two fractions by multiplying the numerator and denominator of the first term by sinx, and the numerator and denominator of the second term by cosx. This gives us:
(1)/(sin²x/cosx+cos²x/sinx).
Simplifying further, we can multiply the numerator and denominator by cosx*sinx to get:
cosx*sinx/((sin²x)+(cos²x)).
Using the trigonometric identity sin²x+cos²x=1, the expression simplifies to:
cosx*sinx/1.
Therefore, the complex fraction (1)/(tanx+cotx) is equal to sinx*cosx.