Final answer:
To verify the identity (sin(x)+tan(x))/(1+sec(x))=sin(x), we can simplify the left-hand side using trigonometric identities. First, express tan(x) as sin(x)/cos(x). Next, combine the two terms in the numerator by finding a common denominator. Finally, simplify the expression further by factoring out sin(x) from the numerator.
Step-by-step explanation:
To verify the identity (sin(x)+tan(x))/(1+sec(x))=sin(x), we can simplify the left-hand side using trigonometric identities.
First, express tan(x) as sin(x)/cos(x). This gives us (sin(x)+(sin(x)/cos(x)))/(1+1/cos(x)). Next, combine the two terms in the numerator by finding a common denominator, which is cos(x). This yields (sin(x)*cos(x)+sin(x))/cos(x).
Finally, simplify the expression further by factoring out sin(x) from the numerator, resulting in sin(x)(cos(x)+1)/cos(x). Since sin(x)/cos(x) is equal to tan(x), we can rewrite the expression as tan(x)+1/cos(x), which is equivalent to sin(x) according to the identity tan(x)+1/cos(x)=sin(x).The mathematical question asks to verify the identity (sin(x)+tan(x))/(1+sec(x))=sin(x).
We start by using trigonometric identities such as tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x).
Substituting these into the equation, we get:
(sin(x) + sin(x)/cos(x))/(1+1/cos(x)) = sin(x) * (1+cos(x))/(cos(x)+1) = sin(x)
This confirms the identity is correct because the terms cancel out appropriately.