Prove the identity #(1/sinx - 1/tanx)^2 -= (1-cosx)/(1+cosx).# ?

Question

Prove the identity #(1/sinx - 1/tanx)^2 -= (1-cosx)/(1+cosx).#

HIX Tutor · Accepted Answer

undefined Sure! Let's look at the identity step by step. We want to prove:

$$
\left(\frac{1}{\sin x} - \frac{1}{\tan x}\right)^2 = \frac{1 - \cos x}{1 + \cos x}
$$

### Step 1: Simplify the left side
First, let’s simplify the left side:

$$
\frac{1}{\sin x} - \frac{1}{\tan x}
$$

What does $\tan x$ equal? Can you remind me?

Anna Andrews · Answer

I would start with the left hand side, by rewriting in terms of sine and cosine.

LHS:

#(1/sinx - 1/(sinx/cosx))^2#

#(1/sinx - cosx/sinx)^2#

#((1 - cosx)/sinx)^2#

#(1 -cosx)^2/sin^2x#

Recall that

#sin^2x +cos^2x = 1 -> sin^2x= 1- cos^2x#.

#(1 - cosx)^2/(1 - cos^2x)#

Now do a little factoring.

#((1 - cosx)(1 - cosx))/((1 + cosx)(1 - cosx))#

#(1 - cosx)/(1 + cosx)#

We now see that LHS = RHS, therefore we've proven this identity.

Hopefully this helps!

HIX Tutor · Answer

undefined Sure! Let's break it down step by step.

1. **Start with the left side:** $(\frac{1}{\sin x} - \frac{1}{\tan x})^2$

2. **Rewrite $\tan x$:** We know that $\tan x = \frac{\sin x}{\cos x}$.

3. **Substitute for $\tan x$:** 
   $\frac{1}{\tan x} = \frac{\cos x}{\sin x}$.

4. **Now, rewrite the expression:**
   $$
   \frac{1}{\sin x} - \frac{\cos x}{\sin x} = \frac{1 - \cos x}{\sin x}.
   $$

5. **Square it:**
   $$
   \left(\frac{1 - \cos x}{\sin x}\right)^2 = \frac{(1 - \cos x)^2}{\sin^2 x}.
   $$

6. **Now, note that $\sin^2 x = 1 - \cos^2 x$:** 
   Therefore,
   $$
   \sin^2 x = (1 - \cos x)(1 + \cos x).
   $$

7. **Replace $\sin^2 x$ in the expression:**
   $$
   \frac{(1 - \cos x)^2}{(1 - \cos x)(1 + \cos x)} = \frac{1 - \cos x}{1 + \cos x}.
   $$

8. **Now, the left side looks like this:**
   $$
   \frac{1 - \cos x}{1 + \cos x}.
   $$

9. **This matches the right side!** The right side is also:
   $$
   \frac{1 - \cos x}{1 + \cos x}.
   $$

Thus, we proved that 
$$
\left(\frac{1}{\sin x} - \frac{1}{\tan x}\right)^2 = \frac{1 - \cos x}{1 + \cos x}.
$$

Do you understand each step?