Hey guys, For those who are fans of pure maths, help me prove the following identity cosx/ tanx(1-sinx) = 1 + 1/sinx I tested it through a calculator and they match so its only the theoretical matching that I cannot do. things you can use are: sin 2 x + cos 2 x = 1 and tanx = sinx/cosx Happy headaches!
Oh all right. One clarification needed, is the (1-sinx) part of the denominator, or is the whole fraction (cosx/tanx) multiplied by it? Here you go: cosx/[(sinx(1-sinx))/cosx] = sinx/sinx + 1/sinx cos^2(x)/[sinx(1-sinx)] = (1+sinx)/sinx ///multiply all by sinx(1-sinx) cos^2x = (1+sinx)(1-sinx) = 1 - sin^2x. Which is true.
uhuh so can u prove it from one side only so that the multiplication of the denominator doesnt come into place? Its a different approach where u consider both sides at once. Its for a boy who has his exams on monday.
cosx/[(sinx(1-sinx))/cosx] = cos^2x/[sinx(1-sinx)] = = (1-sin^2x)/[sinx(1-sinx)] =///using the equivalence that a^2-b^2 = (a+b)(a-b) /// = = [(1+sinx)(1-sinx)]/[sinx(1-sinx)] = =(1+sinx)/sinx = = 1/sinx + sinx/sinx = = 1 + 1/sinx
Another strategy for proving something like that is to separate all the 'complex' looking fractions to ONE side of the equation, so that the other side contains only a 'simple' term. In general, it's easier to reduce a complex formula to something simpler, than it's to try and match one complex formula with another complex formula. so for example, move the last term 1/sinx to the left side, and solve for cosx/[tanx(1-sinx)] - 1/sinx If you get 1, you're done.
Sorry, I just meant that for an equation such as: cosx/tanx(1-sinx) = 1 + 1/sinx ...there are fractions on both sides of "=". Here, both "cosx/tanx(1-sinx)" and "1/sinx" are fractions. If you can isolate these fractions to the left side of "=", the right side of "=" is left with the integer "1": cosx/tanx(1-sinx) = 1 + 1/sinx (1) cosx/tanx(1-sinx) - 1/sinx = 1 (2) To prove (1) is equivalent to proving (2), to prove (2) you can simply reduce "cosx/tanx(1-sinx) - 1/sinx" to "1". I think a student would find it relatively easier if they know the "answer" is "1", and all they have to do is "solve" it.