Question

How do you prove: `sinx/cosx + cosx/sinx = 1`?

Answer 1

It's impossible to prove since this is not true.

Explanation:

You can't prove this, it's not an identity.

Let me show you why.

First of all, you should find the least common multiple to add the two fractions. Your least common multiple is `cos x * sin x`:

`sin x / cos x + cos x / sin x = 1`

`(sin x * sin x ) / (cos x * sin x ) + (cos x * cos x) / (cos x * sin x ) = 1`

`(sin^2x + cos^2 x) / (cos x sin x ) = 1`

Remember that `sin^2 x + cos^2 x = 1`...

`1 / (cos x sin x ) = 1`

Now, this can only be true if the denominator is equal to `1` which would mean that `sin x = 1 / cos x` for all `x`.

As

`1 / cos x = sec x`,

and `sec x` is certainly not the same as `sin x`, your equation can't be an identity.

As you can see from the graph below, the equation doesn't even have any solutions for any `x`:

Graph of `(cos x * sin x - 1)`:

graph{cos x * sin x - 1 [-6.24, 6.244, -3.12, 3.12]}