Final answer:
To simplify the expression (cosx)/(secx+tanx), replace secx with 1/cosx and tanx with sinx/cosx, obtain a common denominator, and cancel out like terms to get the simplified result, 1/(1 + tanx). This is the simplified form of the original expression.
Step-by-step explanation:
To simplify the trigonometric expression (cosx)/(secx+tanx), we can use the fundamental trigonometric identities. We know that secx is equal to 1/cosx and tanx is equal to sinx/cosx. Thus, we can rewrite the expression in terms of sine and cosine.
Substituting these identities into our expression gives us:
(cosx)/((1/cosx) + (sinx/cosx))
To combine the terms in the denominator, we can get a common denominator, which is cosx, leading to:
(cosx)/((cosx + sinx)/cosx)
Now, the cosx in the numerator and cosx in the denominator cancel each other out:
1/(1 + (sinx/cosx))
Since sinx/cosx is tanx, we have finally:
1/(1 + tanx)
This is the simplified form of the original expression.