Final answer:
To simplify the expression 1/(1+sinX), multiply the numerator and denominator by the conjugate of the denominator (1-sinX). Then use the identity sin^2(X) + cos^2(X) = 1 to simplify.
Step-by-step explanation:
To simplify the expression 1/(1+sinX) using trigonometric identities, we can start by multiplying the numerator and denominator by the conjugate of the denominator, which is 1-sinX.
This gives us:
1/(1+sinX) * (1-sinX)/(1-sinX)
Expanding and simplifying further:
= (1 - sinX)/(1 - sin^2(X))
= (1 - sinX)/(cos^2(X))
Finally, using the identity sin^2(X) + cos^2(X) = 1, we can simplify the expression to:
= (1 - sinX)/(1 - sin^2(X))
= 1/cosX
Therefore, the simplified form of the expression 1/(1+sinX) is cosX.