How can you verify #sinx/(sinx+cosx)=(cotx-1)/(cotx+1)# by only manipulating the left side?

Question

Asher Friedman · Accepted Answer

It cannot be done, because it is not an identity. The equation is only true at specific values of x:

#x = tan^-1(-2+-sqrt5)+ 2pin; n in ZZ#

Delilah Aguilar · Answer

undefined To verify the identity sin(x) / (sin(x) + cos(x)) = (cot(x) - 1) / (cot(x) + 1) by only manipulating the left side, we can start with the left side of the equation and manipulate it algebraically until it equals the right side.

Starting with the left side: 
sin(x) / (sin(x) + cos(x))

First, multiply the numerator and denominator by (1 / sin(x) - cos(x)):
(sin(x) * (1 / sin(x) - cos(x))) / ((sin(x) + cos(x)) * (1 / sin(x) - cos(x)))

Now, distribute sin(x) into the numerator:
(1 - sin(x) * cos(x)) / ((sin(x) + cos(x)) * (1 / sin(x) - cos(x)))

Using the reciprocal identities sin(x) = 1 / csc(x) and cos(x) = 1 / sec(x):
(1 - (1 / csc(x)) * (1 / sec(x))) / (((1 / csc(x)) + (1 / sec(x))) * (1 / sin(x) - (1 / sec(x))))

Simplify the terms:
(1 - (1 / (1 / sin(x)))) / (((1 / (1 / sin(x))) + (1 / (1 / cos(x)))) * (1 / sin(x) - (1 / cos(x))))

This simplifies to:
(1 - sin(x)) / ((sin(x) + cos(x)) * (1 - sin(x)))

Now, multiply the numerator and denominator by (-1):
(-1 * (1 - sin(x))) / (-1 * (sin(x) + cos(x)) * (1 - sin(x)))

Simplify:
(sin(x) - 1) / ((sin(x) + cos(x)) * (sin(x) - 1))

Now, we notice that (sin(x) - 1) is the negative of (1 - sin(x)), so they cancel each other out:
= ((1 - sin(x)) / ((sin(x) + cos(x)) * (1 - sin(x))))

Now, we have the right side of the identity:
= (cot(x) - 1) / (cot(x) + 1)

Therefore, the left side equals the right side, and the identity is verified.