Number six, if tangent of x plus 1 over the tangent of x is equal to pi, then what is the product of sine of x times cosine of x equal to? It is not clear how to begin this problem, but we have an equation we can work with. On the right hand side is pi, we don't really know how to deal with it, but on the left hand side we have an expression that can be simplified. So let's simplify it. Let's take tangent of x plus 1 over the tangent of x and write it in terms of sines and cosines. So tangent being sine divided by cosine, its reciprocal would be 1 over the tangent of x or cosine divided by sine of x. And let's add these two fractions. The common denominator would be sine x times cosine of x, which is the expression we are looking for. So this leads us to believe we're on the right path. And in the numerator, by cross-multiplying, we have a sine squared plus cosine squared, which is a well-known identity. Sine squared plus cosine squared is always equal to 1. So we have that pi is equal to 1 over sine times cosine. And taking a reciprocal of both sides of this equation, you see that 1 over pi is equal to the product sine of x times cosine of x. So that is what we were looking for, and the answer is E.

tangent

trigonometry

sine

cosine

identity