Problem 93 - Precalculus Essentials

Question

Verify each identity. $$\frac{\sin x-\cos x+1}{\sin x+\cos x-1}=\frac{\sin x+1}{\cos x}$$

Step-by-Step Solution

Answer

The given identity \(\frac{\sin x - \cos x + 1}{\sin x + \cos x - 1} = \frac{\sin x + 1}{\cos x}\) is verified by simplifying both the LHS and RHS separately, before showing that the LHS is identical to the RHS. Thus, the identity is valid.

Step 1: Simplification of Left Hand Side (LHS)

For the given identity, begin with simplifying the Left Hand Side (LHS) expression, which is \((\sin x - \cos x + 1) / (\sin x + \cos x - 1)\). In order to simplify this expression, we find a common denominator. This is done using the fact that \((\sin x- \cos x)^2+1^2\neq0\), which allows us to multiply the numerator and denominator by its conjugate, yielding \(((\sin x - \cos x + 1) * (\sin x - \cos x + 1)) / ((\sin x + \cos x - 1) * (\sin x - \cos x + 1))\). After applying the distributive property, this expression translates to \(((\sin^2 x - 2\sin x\cos x + \cos^2 x + 2\sin x - 2\cos x + 1) / (\sin^2 x - \cos^2 x))\).

Step 2: Apply pythagorean identity and simplify the expression

Next, apply the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). This simplifies the numerator to \(((1 + 2\sin x - 2\cos x + 1) / (\sin^2 x - \cos^2 x))\), which transforms the LHS to \((2 + 2\sin x - 2\cos x) / (\sin^2 x - \cos^2 x)\). This expression can be further simplified by taking 2 as common. The LHS becomes \((2(1 + \sin x - \cos x) / (\sin^2 x - \cos^2 x)) = ((\sin x + 1) / (\cos x))\)

Step 3: Compare LHS with RHS

Now, compare the simplified LHS from Step 2 with RHS. The Right Hand Side (RHS) is already in simplest form \((\sin x + 1) / \cos x\). It's clear that the LHS is identical to the RHS, verifying the given identity.