There are two questions of my homework I'm having trouble with.. I think we are supposed to show how they are true. In other words, make one side look exactly like the other one by using the identities.
1. cosX/(1-sinX)=(1+sinX)/cosX
2. (secX+tanX)/(secX-tanX)=(1+2sinX+sin^(2)X)/cos^(2)X)
Please help, thanks
for the first, multiply the LS by (1+sinx)/(1+sinx)
It will turn into the RS
2.
LS = (1/cosx + sinx/cosx)/(1/cosx - sinx/cosx)
we have a common denominator
and remember how to divide fractions ...
= (1+sinx)/(1-sinx)
RS = (1+sinx)(1+sinx)/(1 - sin^2 x)
= (1+sinx)(1+sinx)/[(1+sinx)(1+sinx)]
= (1+sinx)/(1-sinx)
= LS
Thanks!
Sure, I can help you with those questions. To prove that the given equations are true, we need to simplify each side of the equation using trigonometric identities until they are equivalent. Let's start with the first equation:
1. cosX / (1 - sinX) = (1 + sinX) / cosX
To simplify the left side, we will multiply both the numerator and denominator by (1 + sinX):
cosX * (1 + sinX) / [(1 - sinX) * (1 + sinX)]
Expanding the denominator:
cosX * (1 + sinX) / (1 - sin^2X)
Using the identity sin^2X + cos^2X = 1:
cosX * (1 + sinX) / cos^2X
Now, let's simplify the right side of the equation:
(1 + sinX) / cosX
Multiplying numerator and denominator by (1 - sinX):
(1 + sinX) * (1 - sinX) / (cosX * (1 - sinX))
Expanding the numerator:
(1 - sin^2X) / (cosX * (1 - sinX))
Using the identity sin^2X + cos^2X = 1:
(cos^2X) / (cosX * (1 - sinX))
Canceling out the common term cosX, we get:
cosX / (1 - sinX)
As you can see, both sides of the equation simplify to the same expression, so the equation is true.
Now, let's move on to the second equation:
2. (secX + tanX) / (secX - tanX) = (1 + 2sinX + sin^2X) / cos^2X
To simplify both sides, we need to work with the left side first:
(secX + tanX) / (secX - tanX)
To make it easier, let's multiply both the numerator and denominator by (secX + tanX):
(secX + tanX) * (secX + tanX) / [(secX - tanX) * (secX + tanX)]
Expanding both the numerator and denominator:
(sec^2X + 2secXtanX + tan^2X) / (sec^2X - tan^2X)
Using the identity 1 + tan^2X = sec^2X:
(sec^2X + 2secXtanX + 1) / (sec^2X - tan^2X)
Again, using the identity 1 - tan^2X = sec^2X:
(sec^2X + 2secXtanX + 1) / (1)
Simplifying further, we get:
sec^2X + 2secXtanX + 1
Now, let's simplify the right side of the equation:
(1 + 2sinX + sin^2X) / cos^2X
Multiplying numerator and denominator by 1/cos^2X:
(1/cos^2X + 2sinX/cos^2X + sin^2X/cos^2X)
Using the identity sec^2X = 1/cos^2X:
(sec^2X + 2secXtanX + 1) / (1)
As we can see, both sides of the equation simplify to the same expression once again, so the equation is true.
I hope that helps! Let me know if you have any further questions.