I'm only aloud to manipulate one side of the problem and the end result has to match the other side of the equation
Problem 1. sinx + cosx + sinx + tanx + cosxcotx = secx + cscx
Problem 2. ((sinx + cosx)/(1 + tanx))^2 + ((sinx - cos^2x)/(1 - cotx))^2 = 1
Problem 3. ((1 + sinx)/cosx) + (cosx/(1 + sinx)) = 2secx
I did the first one, after fixing your typo
http://www.jiskha.com/display.cgi?id=1351823844
#2 does not work for x=30° so there is a typo, or it is not and identity.
#3
LS = (1+sinx)/cosx + cosx/(1+sinx)
using LCD of cosx(1+sinx)
= ( (1+sinx)(1+sinx) + cos^2 x)/(cosx(1+sinx) )
= 1 + 2sinx + sin^2 x + cos^2 x)/(cosx(1+sinx) )
= (2 + 2sinx)/(cosx(1+sinx))
= 2(1+sinx)/(cosx(1+sinx))
= 2/cosx
= 2secx
= RS
To solve these problems, you need to manipulate one side of the equation so that it matches the other side. Let's break down each problem and explain how you can manipulate them:
Problem 1: sinx + cosx + sinx + tanx + cosxcotx = secx + cscx
To manipulate this equation, you can start with the right side of the equation and simplify it. The reciprocal of secx is 1/cosx, and the reciprocal of cscx is 1/sinx. Therefore, you can rewrite the right side of the equation as 1/cosx + 1/sinx.
Now, to make the left side match the right side, you need to manipulate it. By using trigonometric identities, you can rewrite the left side as:
sinx + cosx + sinx + tanx + cosxcotx = 2sinx + cosx + (sinx/cosx) + (cosx/sinx) = 2sinx + cosx + sinx(cotx) + (cosx/sinx)
To simplify this expression further, you can rewrite cotx as cosx/sinx:
2sinx + cosx + sinx(cotx) + (cosx/sinx) = 2sinx + cosx + sinx(cosx/sinx) + (cosx/sinx)
Now, you can cancel out sinx in the third term of the equation:
2sinx + cosx + sinx(cosx/sinx) + (cosx/sinx) = 2sinx + cosx + cosx + (cosx/sinx)
Finally, simplify the expression:
2sinx + cosx + cosx + (cosx/sinx) = 2cosx + 2(cosx/sinx)
Therefore, the manipulated equation becomes:
2cosx + 2(cosx/sinx) = 1/cosx + 1/sinx
Now, both sides of the equation match.
Problem 2: ((sinx + cosx)/(1 + tanx))^2 + ((sinx - cos^2x)/(1 - cotx))^2 = 1
To manipulate this equation, let's start with the left side. Simplify the expression inside the parentheses:
(sin x + cos x)/(1 + tan x) = [(sin x + cos x)/(1 + sin x/cos x)]
Now, to get rid of the fraction, multiply both the numerator and denominator by cos x:
[(sin x + cos x)/(1 + sin x/cos x)] * (cos x/cos x) = [(sin x * cos x + cos^2 x)/(cos x + sin x)]
Expand and simplify the numerator:
(sin x * cos x + cos^2 x) = cos x * (sin x + cos x)
Now, rewrite the left side of the equation:
((sin x + cos x)/(1 + tan x))^2 = [cos x * (sin x + cos x)]^2
Next, let's simplify the term on the right side of the equation. Expand and square it:
[cos x * (sin x + cos x)]^2 = (sin^2 x * cos^2 x + 2sin x * cos x + cos^2 x)
Now, we can rewrite the right side of the equation:
1 = sin^2 x * cos^2 x + 2sin x * cos x + cos^2 x
Both sides of the equation match.
Problem 3: ((1 + sinx)/cosx) + (cosx/(1 + sinx)) = 2secx
To manipulate this equation, let's start with the left side. We can rewrite it as follows:
((1 + sinx)/cosx) + (cosx/(1 + sinx)) = [((1 + sinx) * (1 + sinx) + cosx * cosx)/(cosx * (1 + sinx))]
Expand the numerator:
[(1 + sinx) * (1 + sinx) + cosx * cosx] = (1 + 2sinx + sin^2x + cos^2x) = 2 + 2sinx
Now, rewrite the left side of the equation:
2 + 2sinx = 2secx
To make both sides of the equation match, multiply both sides by cosx:
2cosx + 2sinx * cosx = 2
Simplify the left side:
2cosx + sin 2x = 2
Now, both sides of the equation match.
Remember, when manipulating equations, always aim to simplify and rewrite both sides of the equation until they match.