Question

Cos y/ 1-sin y= 1+sin y/cos y. Verify the identity. Show All Steps!

Answer 1

Answer:

When proving identities, the answer is in the explanation.

Step-by-step explanation:

$\frac{\cos(y)}{1-\sin(y)}$

I have two terms in this denominator here.

I also know that $1-\sin^2(\theta)=\cos^2(theta)$ by Pythagorean Identity.  

So I don't know how comfortable you are with multiplying this denominator's conjugate on top and bottom here but that is exactly what I would do here.  There will be other problems will you have to do this.

$\frac{\cos(y)}{1-\sin(y)} \cdot \frac{1+\sin(y)}{1+\sin(y)}$

Big note here: When multiplying conjugates all you have to do is multiply fist and last.  You do not need to do the whole foil.  That is when you are multiplying something like $(a-b)(a+b)$, the result is just $a^2-b^2$.

Let's do that here with our problem in the denominator.

$\frac{\cos(y)}{1-\sin(y)} \cdot \frac{1+\sin(y)}{1+\sin(y)}$

$\frac{\cos(y)(1+\sin(y))}{(1-\sin(y))(1+\sin(y)}$

$\frac{\cos(y)(1+\sin(y))}{1^2-\sin^2(y)}$

$\frac{\cos(y)(1+\sin(y))}{1-\sin^2(y)}$

$\frac{\cos(y)(1+\sin(y))}{cos^2(y)}$

In that last step, I apply the Pythagorean Identity I mentioned way above.

Now You have a factor of cos(y) on top and bottom, so you can cancel them out.  What we are really saying is that cos(y)/cos(y)=1.

$\frac{1+\sin(y)}{cos(y)}$

This is the desired result.

We are done.