Final answer:
After simplifying the right side of the equation using the Pythagorean identity, we find that the original identity sin(x)tan(x) = 1 + cos^2(x)/cos(x) is false, and the correct equation is sin(x)tan(x) = sin^2(x)/cos(x).
Step-by-step explanation:
To prove or disprove the identity sinxtanx = 1 + cos^2x/cosx, we will manipulate the left side of the equation and see if it simplifies to the right side. Starting with the left side, we have sinxtanx = sinx(sinx/cosx). Using the identity tanx = sinx/cosx, we can simplify further to sinx(sinx/(sinx/cosx)). Simplifying the expression inside the parentheses gives us sin^2x/cosx. Factoring out sin^2x from the numerator, we have sinx(sin^2x/cosx). Finally, using the identity sin^2x = 1 - cos^2x, we get sinx(1 - cos^2x/cosx), which simplifies to sinx - cos^2x. Therefore, the identity sinxtanx = 1 + cos^2x/cosx is false, and the correct equation is sinxtanx = sinx - cos^2x.
The student asked to prove or disprove the identity sin(x)tan(x) = 1 + cos2(x)/cos(x). To assess the identity, let us first simplify the right side by incorporating the Pythagorean identity sin2(x) + cos2(x) = 1. The thus becomes sin(x)sin(x)/cos(x) = sin2(x)/cos(x). We will substitute this into our original equation and see that the left side sin(x)tan(x) simplifies to sin2(x)/cos(x), which shows that the identity is false. To correct the equation, we replace the right side to get sin(x)tan(x) = sin2(x)/cos(x).