SOLUTION: Prove each identity: tan^2(x)/1+tan^2(x) =sin^2(x) and sin^2(x)(1+1/tan^2(x))=1 i get confused for example sin^2(x) and (sinx)^2

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Question 1202859: Prove each identity:
tan^2(x)/1+tan^2(x) =sin^2(x)
and
sin^2(x)(1+1/tan^2(x))=1
i get confused for example sin^2(x) and (sinx)^2
Found 4 solutions by josgarithmetic, math_tutor2020, greenestamps, MathTherapy:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Maybe you mean for your first one to be
tan^2(x)/(1+tan^2(x)) =sin^2(x)

or in the rendering tags
tan%5E2%28x%29%2F%281%2Btan%5E2%28x%29%29+=sin%5E2%28x%29
.

If you know or can find some specific identities you may see as possible step to begin
%28tan%5E2%28x%29%29%2F%28sec%5E2%28x%29%29=sin%5E2%28x%29
and then maybe you can continue to verify the rest of the way.


----
----
sin%5E2%28x%29%281%2B1%2F%28sin%5E2%28x%29%2Fcos%5E2%28x%29%29%29=1
sin%5E2%28x%29%281%2B%28cos%5E2%28x%29%29%2F%28sin%5E2%28x%29%29%29=1


Distributive Property:
sin%5E2%28x%29%2Bsin%5E2%28x%29%28%28cos%5E2%28x%29%29%2F%28sin%5E2%28x%29%29%29=1
sin%5E2%28x%29%2Bcos%5E2%28x%29=1 IDENTITY; proved

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

I'll prove the 1st identity and leave the other identity for the student to do.

LHS = left hand side
RHS = right hand side

I will transform the LHS into the RHS.
The RHS will be kept the same.

%28tan%5E2%28x%29%29%2F%281+%2B+tan%5E2%28x%29%29+=+sin%5E2%28x%29

%28tan%5E2%28x%29%29%2F%28sec%5E2%28x%29%29+=+sin%5E2%28x%29 Use the identity 1+tan^2 = sec^2

matrix%281%2C3%2Ctan%5E2%28x%29%2C%22divide%22%2Csec%5E2%28x%29%29+=+sin%5E2%28x%29

Use identities tan = sin/cos and sec = 1/cos.







%28%28sin%5E2%28x%29%29%2F1%29%2A%281%2F1%29+=+sin%5E2%28x%29

sin%5E2%28x%29+=+sin%5E2%28x%29

The identity has been confirmed.

Answer by greenestamps(13215) About Me  (Show Source):
You can put this solution on YOUR website!


Preliminary comments....

(1) "sin^2(x)" and (sinx)^2 are both used to represent the square of sin(x).

(2) Use parentheses properly. The first equation as you show it is not an identity:

tan^2(x)/1+tan^2(x) =sin^2(x) ---> tan%5E2%28x%29%2F1%2Btan%5E2%28x%29=sin%5E2%28x%29

The equation you intended to show is

tan^2(x)/(1+tan^2(x)) =sin^2(x) ---> tan%5E2%28x%29%2F%281%2Btan%5E2%28x%29%29=sin%5E2%28x%29

Now my approaches to these....

Both of the other tutors used the identity 1+tan^2(x) = sec^2(x). That is certainly one way to start. But after that they turn everything into sines and cosines, so it seems easiest just to do that at the beginning.

(a) tan%5E2%28x%29%2F%281%2Btan%5E2%28x%29%29

%28sin%5E2%28x%29%2Fcos%5E2%28x%29%29%2F%281%2Bsin%5E2%28x%29%2Fcos%5E2%28x%29%29



%28sin%5E2%28x%29%2Fcos%5E2%28x%29%29%2F%281%2Fcos%5E2%28x%29%29

sin%5E2%28x%29

(b) sin%5E2%28x%29%281%2B1%2Ftan%5E2%28x%29%29

sin%5E2%28x%29%281%2B%28cos%5E2%28x%29%2Fsin%5E2%28x%29%29%29

sin%5E2%28x%29%2Bcos%5E2%28x%29=1

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!
Prove each identity:

tan^2(x)/1+tan^2(x) =sin^2(x)

and 

sin^2(x)(1+1/tan^2(x))=1

i get confused for example sin^2(x) and (sinx)^2

Proving the left side equal to the right side!
            
                  ------ Substituting matrix%281%2C3%2C+sec%5E2+%28x%29%2C+for%2C+1+%2B+tan%5E2+%28x%29%29
              ------ Substituting 
     sin%5E2+%28x%29%2Fcos%5E2+%28x%29 รท matrix%281%2C3%2C+1%2Fcos%5E2+%28x%29%2C+%22=%22%2C+highlight%28sin%5E2%28x%29%29%29