Hi! Okay so we just learned this last week and I completely lost on what to do. These are some warm up problems we did in class but I still don't know how to do them? Can someone help me? Thanks! :)
Directions: Verify the idenity for each of the problems. Show all work!
1.)cot^2y(sec^2-y-1)=1
2.) cosx-cosx/1-tanx=sin xcoxc/sinx-cosx
If you have a typo in #1 and you meant:
cot^2y(sec^2 y-1)=1 , then
LS = cot^2 y (tan^2 y)
= 1
= RS
recall that :
tanx = sinx/cosx and cotx = cosx/sinx
so tanxcotx = (sinx/cosx)(cosx/sinx) = 1
The first substitution is one of the basic Pythagorean identities derived from
sin^2 x + cos^2 x = 1, dividing each term by cos^2 x
#2. Again, I think you have a typo (no brackets)
and it could be
cosx - cosx/(1-tanx) = ...
or on the right side: sinxcosx/(sinx - cosx)
fix the equation with proper brackets and I will attempt it
Oh, yes, I apologize for not including those brackets! I wrote that wrong. The right way is:
cosx - cosx/(1-tanx) = sinxcosx/(sinx - cosx)
cosx - cosx/(1-tanx) = sinxcosx/(sinx - cosx)
LS = cosx - cosx/(1 - sinx/cosx)
= cosx - cosx/( (cosx - sinx)/cosx )
= cosx - cosx(cosx)/(cosx - sinx)
= cosx + cos^2 x/(sinx - cosx)
= (sinxcosx - cos^2 x + cos^2 x)/(sinx - cosx)
= sinxcosx/(sinx - cosx)
= RS
Of course, I can help you with these problems! Let's start by breaking down each problem and explaining the steps to solve them.
1.) cot^2y(sec^2-y-1)=1
To verify the identity in this problem, we'll manipulate the left side of the equation until it simplifies to the right side.
Step 1: Rewrite cot^2y as 1/(tan^2y) since cotangent is the reciprocal of tangent.
So the equation becomes:
1/(tan^2y)(sec^2y - 1) = 1
Step 2: Expand sec^2y - 1. The identity for sec^2y - 1 is tan^2y.
So the equation simplifies to:
1/(tan^2y)(tan^2y) = 1
Step 3: Simplify the expression on the left side by canceling out the common term.
This simplifies to:
1 = 1
Since the left side of the equation is equal to the right side, the identity is verified.
2.) cosx - cosx/1 - tanx = sinx cosx/sinx - cosx
Again, let's manipulate the left side of the equation to simplify it.
Step 1: Rewrite cosx - cosx as 0.
The equation becomes:
0/1 - tanx = sinx cosx/sinx - cosx
Step 2: Simplify the expression on the left side.
0 - tanx = sinx cosx/sinx - cosx
Step 3: Simplify the expression on the right side by canceling out the common term.
0 - tanx = cosx
Step 4: Simplify the expression on the left side by subtracting tanx from 0.
- tanx = cosx
Now, this is not the same as the expression on the right side, so the given equation does not hold true. Therefore, the identity is not verified.
Remember, when verifying identities, you need to manipulate the expression on one side of the equation to transform it into the same form as the expression on the other side. If the two sides of the equation become identical, the identity is verified. If they do not become identical, the identity is not verified.
I hope this explanation helps! If you have any more questions, feel free to ask.