Prove the following identity:
1/tanx + tanx = 1/sinxcosx
I can't seem to prove it. This is my work, I must've made a mistake somewhere:
Converted 1/tanx: 1/sinx/cosx + sinx/cosx = 1/sinxcosx
Simplified 1/sinx/cosx: cosx/sinx + sinx/cosx = 1/sinxcosx
Found common denominator and multiplied numerator: sinxcosx/sinxcosx + sinxcosx/sinxcosx = 1/sinxcosx
Simplified: 2sinxcosx/sinxcosx
Wouldn't that simplify to just sinxcosx, not 1/sinxcosx?
cosx/sinx + sinx/cosx = 1/sinxcosx
common denominator: sinxcosx
(cos^2x+sin^2x)/sinxcosx
not it is proved, as the numberator is 1.
Let's take a closer look at your work to identify the mistake.
You correctly converted 1/tanx to sinx/cosx in the first step, but there seems to be a mistake in the next step.
When you add the two fractions (cosx/sinx + sinx/cosx), you need to find a common denominator. In this case, the common denominator is sinx*cosx. To adjust the first fraction, you multiply both the numerator and denominator by cosx. Similarly, for the second fraction, you multiply both the numerator and denominator by sinx.
The correct step would be:
(cosx * cosx)/(sinx * cosx) + (sinx * sinx)/(sinx * cosx)
Simplifying further:
cos^2x + sin^2x / (sinx * cosx)
Now, we know that sin^2x + cos^2x = 1 (from the Pythagorean identity), so we can replace this expression in the numerator:
1 / (sinx * cosx)
Therefore, the expression 1/tanx + tanx simplifies to 1/(sinx * cosx), proving the given identity.