Hi, I am a senior in High School having a really difficult time with two problems. I have to prove using the trigonometric identities that they equal each other but I am having a really hard time trying to get them to equal each other. I've spent over 2 hours trying to get these two problems right but I can't. I don't know if the math problems themselves are working because there is some error or that I am just doing them wrong (even though I've spent a long time trying various methods to get them correct). So I was wondering if someone could help me really soon. Here are the problems:
Thank you so much!
1: (1+Cotx)^2/Tanx = CosxCsc^3x + 2Cot^2x
2: Cosx/1-Sinx - Tanx = Secx
#2:
Here are some basic trig identities to help with this problem.
sin^2x + cos^2x = 1
secx = 1/cosx
tanx = sinx/cosx
Let's try to get the left side to look like the right side.
cosx/(1-sinx) - tanx =
cosx/(1-sinx) - sinx/cosx =
Common denominator is: (1-sinx)(cosx)
Therefore:
cos^2x/(1-sinx)(cosx) - sinx(1-sinx)/(1-sinx)(cosx) =
[cos^2x - (sinx - sin^2x)]/(1-sinx)(cosx) =
(cos^2x + sin^2x - sinx)/(1-sinx)(cosx) =
(1 - sinx)/(1-sinx)(cosx) =
1/cosx =
secx = secx
I hope this helps with this one; perhaps someone else can help with the first problem.
(1+cotx)^2/tanx
=(1+2cotx+cot^2x)/(sinx/cosx)
=[(1+cot^2x)+2cotx]/(sinx/cosx)
=csc^2x/(sinx/cosx)+2cotx/(sinx/cosx)
=csc^2xcosx/(1/sinx)+2cotxcosx/sinx
=csc63xcosx+2cot^2x
For problem #1, let's work through it step by step. Here are some basic trigonometric identities that will be helpful:
1. Pythagorean identity: sin^2(x) + cos^2(x) = 1
2. reciprocal identities: csc(x) = 1/sin(x), cot(x) = 1/tan(x), sec(x) = 1/cos(x)
3. quotient identities: tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x)
Now, let's start with the left side of the equation and try to simplify it.
(1 + cot(x))^2 / tan(x) =
(1 + cot(x)) * (1 + cot(x)) / tan(x) =
(1 + cot(x)) * (1 + (cos(x)/sin(x))) / (sin(x)/cos(x)) =
(1 + cos(x)/sin(x)) * (1 + cos(x)/sin(x)) * (cos(x)/sin(x)) =
[(sin(x) + cos(x))/sin(x)] * [(sin(x) + cos(x))/sin(x)] * (cos(x)/sin(x)) =
Now, let's simplify the numerator:
(sin(x) + cos(x))^2 =
sin^2(x) + 2sin(x)cos(x) + cos^2(x) =
(1 - cos^2(x)) + 2sin(x)cos(x) + cos^2(x) [using Pythagorean identity] =
1 - cos^2(x) + 2sin(x)cos(x) + cos^2(x) =
1 + 2sin(x)cos(x)
Now, let's simplify the denominator:
sin(x)/cos(x) =
1/cos(x) =
sec(x)
Bringing it all together:
(1 + cot(x))^2 / tan(x) =
[(sin(x) + cos(x))/sin(x)] * [(sin(x) + cos(x))/sin(x)] * (cos(x)/sin(x)) =
(1 + 2sin(x)cos(x))/sin(x) * cos(x)/sin(x) =
1 + 2sin(x)cos(x) =
1 + 2sin(x)cos(x) =
1 + cos(x)sin(x) =
cos(x)sin(x) + 1
Therefore, we have:
(1 + cot(x))^2 / tan(x) = cos(x)sin(x) + 1
I hope this helps you solve the problem! Let me know if you have any further questions.