Final answer:
The correct first step in simplifying the expression tanx/(1 + secx) is option A, where we multiply by the conjugate to simplify the denominator.
Step-by-step explanation:
The simplification of the expression tanx/(1 + secx) can be tackled by using trigonometric identities to transform the expression into a simpler form. Let's consider the given options:
• A. tanx(1 - secx) / (1 + secx)(1 - secx) is the application of the multiplication by the conjugate. This is a common technique used to simplify expressions involving secant or other trigonometric functions.
• B. tanx + sinx does not appear to simplify the original expression.
• C. tanx.cosx / cosx + secx isn’t a simplification as it introduces extra terms instead of reducing them.
• D. None of the above suggests that none of the given options provide a valid simplification step.
Option A is the valid choice here. It employs the technique of multiplying by the conjugate (1 - secx), which is a technique often used to get rid of complex denominators in trigonometric expressions. This works because (1 + secx)(1 - secx) becomes 1 - sec2x, which can be further simplified using the identity sec2x = 1 + tan2x, leading to a simpler expression. Therefore, the acceptable first step in simplifying the expression tanx/(1 + secx) would be option A: tanx(1 - secx) / (1 + secx)(1 - secx).