Help with Establishing Identities

[berge0146]

Hoping someone could help me out with establishing these trig identities in a step by step process because I'm just not understanding them. There are three problems I understand how to factor them all to sin and cos, I guess I just want to compare methods to make sure I'm doing all the steps correctly because I don't think I am. Thanks in advance for the help.

1.) sin(x)[(cot(x)+tan(x))]=sec(x)


2.) 1 - [sin^2(x)/(1-cos(x))]=-cos(x)


3.) [(1+sin(x))/(1-sin(x))] - [(1-sin(x))/(1+sin(x))]= 4tan(x)sec(x)

 

[srmichael]

Hoping someone could help me out with establishing these trig identities in a step by step process because I'm just not understanding them. There are three problems I understand how to factor them all to sin and cos, I guess I just want to compare methods to make sure I'm doing all the steps correctly because I don't think I am. Thanks in advance for the help.

1.) sin(x)[(cot(x)+tan(x))]=sec(x)


2.) 1 - [sin^2(x)/(1-cos(x))]=-cos(x)


3.) [(1+sin(x))/(1-sin(x))] - [(1-sin(x))/(1+sin(x))]= 4tan(x)sec(x)

Identities alwyas give students problems mainly because there is not just one way to approach the verification of an identity. Here are some hints for each of the ones above:

1) Change everything to sin and cos

2) sin^2(x) = 1-cos^2(x) and this is the difference of two squares

3) Combine the two fractions into one fraction by getting a common denominator

 

[Deleted member 4993]

Hoping someone could help me out with establishing these trig identities in a step by step process because I'm just not understanding them. There are three problems I understand how to factor them all to sin and cos, I guess I just want to compare methods to make sure I'm doing all the steps correctly because I don't think I am. Thanks in advance for the help.

1.) sin(x)[(cot(x)+tan(x))]=sec(x)

LHS = sin(x)*cot(x) + sin(x)*tan(x)

= cos(x) + sin²(x)/cos(x)

= [cos²(x) + sin²(x)]/cos(x)

= 1/cos(x) = sec(x) = RHS





2.) 1 - [sin^2(x)/(1-cos(x))]=-cos(x)


3.) [(1+sin(x))/(1-sin(x))] - [(1-sin(x))/(1+sin(x))]= 4tan(x)sec(x)

.

 

[berge0146]

Hey thanks for the help first off. I really understand the first two now but I am still having trouble with the third. When I put them into one fraction I get

sin^2(x)/[(1-sin(x))(1+sin(x))] which should be sin^2(x)/(-cos^2(x)) I think I am misunderstanding because this is where im getting confused because this just don't seem right and this problem is starting to bug me now :-?. Thanks again for all your help because you guys really helped me understand the first two.

 

[srmichael]

Hey thanks for the help first off. I really understand the first two now but I am still having trouble with the third. When I put them into one fraction I get

sin^2(x)/[(1-sin(x))(1+sin(x))] which should be sin^2(x)/(-cos^2(x)) I think I am misunderstanding because this is where im getting confused because this just don't seem right and this problem is starting to bug me now :-?. Thanks again for all your help because you guys really helped me understand the first two.

Ugh, this will be ugly without LaTex, but here goes:

[(1 + sinx)/(1 - sinx)] - [(1 - sinx)/(1 + sinx)]

[(1 + sinx)^2 - (1 - sinx)^2]/[(1 + sinx)(1 - sinx)]

[(1 + 2sinx + sin^2(x) - (1 - 2sinx + sin^2(x)]/[1 - sin^2(x)]

4sinx/cos^2(x)

4sinx/[cos(x)cos(x)]

4sinx/cos(x) * 1/cos(x)

4tanxsecx

 

[berge0146]

Thats awesome thanks so much. I understood everything you did through the example. I didn't get it at first but once I went through the problem and made sure my answers matched up with yours, it's like I see the light . This will always help me with establishing identities now, even though they have now become my least favorite, but they're not so bad once you start to understand.