How do you solve the following identity sec x cos x equals sin x tan x?

Answer 1

I'm not sure what u mean by solving an identity. An identity is an equation that is always true, i.e., true for all values of the variable(s), in this case x. The only thing u can do with an identity is to prove that it is an identity. The problem here is that it is not an identity. Since sec x = 1 / cos x, the left side of the equation, when defined, is always = 1 (it is not defined when cos x = 0, because u cannot divide by 0). The right side, since tan x = sin x / cos x, is = sin2 x / cos x. When x = 0, the right side = 0 / 1 = 0. So it is not an identity.

Perhaps u meant how do u solve the equation. That is a completely different question, which means find what values of x makes the equation true. In this case, we need to find when the right side is = 1, so sin2 x = cos x (but not 0). Since sin2 x = 1 - cos2 x, this amounts to 1 - cos2 x = cos x, or 1 - cos2 x - cos x = 0. This is a quadratic equation in cos x, whose solutions are cos x = 0.61803398874989484820458683436564 and cos x = -1.6180339887498948482045868343656. The latter never happens (for real x), because cos x is always between -1 and 1. The other happens when x = 51.82729237298775250653169866715 degrees. This turns out to be, I believe, the arc cosine of the golden ratio.

Correction: 0.61803398874989484820458683436564 is actually the reciprocal of the golden ratio, so the answer I gave above is the arc sine of the golden ratio. Also, it is not the only solution. Subtracting it from 360 degrees gives 308.17270762701224749346830133285 degrees, which is another solution. Adding or subtracting any multiple of 360 degrees to or from either of those solutions will also work, since all trig functions are periodic with period 2 pi radians, or 360 degrees.

"Dingbot suspects that this answer contains gibberish". Dingbot is wrong - it looks fine to me.

My only reservation is the excess of precision. Surely four to six significant figures are enough (!). It's probably time to stop slavishly copying answers from Windows Calculator.

You're right. That's where I got the numbers from. I just thought that since this is not a physical measurement but a number that is the solution to a pure math equation, all the digits it gives me (except maybe the last one) are correct, so I might as well use what I have, rather than try to decide how many digits of precision to round to. I'll keep that in mind for future reference.