I need help Proving an Identity

[Whoppie]

I need some guidance on proving these identitles. Any help would be appreciated.

1) csc x cot x/sec x = cot^2x

2) csc A(sin^2 A +cos^2 A tan A)/sin A+cos A =1

 

[galactus]

#1:

\(\displaystyle \L\\\frac{csc(x)cot(x)}{sec(x)}\\=\frac{\frac{1}{sin(x)}\cdot\frac{1}{tan(x)}}{\frac{1}{cos(x)}}\\=\left(\frac{1}{sin(x)}\cdot\frac{1}{tan(x)}\right)cos(x)\\=\frac{cos(x)}{sin(x)}\cdot\frac{1}{tan(x)}\)

Now, see it?.

 

[Whoppie]

Oh my god I didn't even notice that. Because Cos(x)/Sin(x) is Cot(x). I was doing something totally different and I kept getting stuck Thanks alot.

 

[Whoppie]

For the second one would I have to factor out (sin^2 A +cos^2 A tan A) with foil and then start work on it or is there another way?

 

[tkhunny]

What do you mean "factor with foil"? Why not just "factor"? Anyway, don't do that.

csc(x) = 1/sin(x)

tan(x) = sin(x)/cos(x)

That's about all you need. The rest is algebra.

 

[Whoppie]

Sorry I think I said that backwards I shudn't have put foil in there at all

 

[Whoppie]

Ok can I cancel out the (sin^2A+cos^2A) and make it (SinA+CosA)?

 

[tkhunny]

Absolutely not! Try it:

(2^2 + 3^2)/(2 + 3) = (4+9)/5 = 13/5

After your proposed reduction: 2+3 = 5 -- Nope! That's not 13/5.

Why do you still have sine squared and cosine squared? You didn't use your hints.

Simplify and factor the numerator AFTER using both hints.

 

[Whoppie]

Sorry I'm all confused would the Sin^2A+Cos^2A become 1 or Sin^2A+Cos^2x+CosxSinx. Sorry again I always make things harder then they shud be and confuse myself

 

[galactus]

Alas, poor Whoppie, you are making it harder than it needs to be.

Remember, \(\displaystyle cscA=\frac{1}{sinA}\).

\(\displaystyle \L\\\frac{\frac{1}{sinA}(sin^{2}A+cos^{2}AtanA)}{sinA+cosA}\)



\(\displaystyle \L\\\frac{\frac{sin^{2}A}{sinA}+\frac{cos^{2}AtanA}{sinA}}{sinA+cosA}\)

Now, simplify the top and you should see it.

Remember what \(\displaystyle \L\\\frac{tanA}{sinA}\) is equivalent to?.

 

[Whoppie]

Yea I was doing something different and it was making me mad. For some reason I was trying to take the Tan out completely. I don't know when it comes to math I'm stumped for the most part just gotta keep trying I guess.

Thank u both for all your help