Im really struggling with these proving identities problems can somebody please show me how to do these? I'm only aloud to manipulate one side of the equation and it has to equal the other side of the equation at the end
Problem 1. (1-tanx) = (cosx-sinx)/cosx
Problem 2. (secx+tanx)/(cscx-cotx) - (secx-tanx)/(cscx+cotx) = 2(secx+cscx)
a. multipy the left side by cos/cos, you get
(cos-sin)/cos
b. on the left, get a common denominator
(csc^2-ctn^2)
then combine both terms into one fraction. It reduces rapidaly. If you get stuck convert all terms to sin, cos)
Thanks! I feel stupid now that I didn't see that before, but I appreciate your help
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Certainly! I can help you solve proving identities problems.
Problem 1:
To prove the identity (1 - tanx) = (cosx - sinx) / cosx, we will start with the left side and manipulate it to make it equal to the right side.
Step 1:
Starting with the left side, which is (1 - tanx), let's simplify it further.
(1 - tanx) = (1 - sinx/cosx) [Using the definition of tanx]
Step 2:
Now, let's simplify the right side, which is (cosx - sinx) / cosx.
(cosx - sinx) / cosx = cosx / cosx - sinx / cosx [Using the distributive property]
Step 3:
Applying the property of dividing a fraction by the same non-zero number, we have:
cosx / cosx - sinx / cosx = 1 - sinx / cosx
Step 4:
Therefore, we have manipulated the left side to be equal to the right side:
(1 - tanx) = (cosx - sinx) / cosx = 1 - sinx / cosx
Problem 2:
To prove the identity (secx + tanx) / (cscx - cotx) - (secx - tanx) / (cscx + cotx) = 2(secx + cscx), we will manipulate one side of the equation to make it equal to the other side.
Step 1:
Starting with the left side, let's manipulate it further:
(secx + tanx) / (cscx - cotx) - (secx - tanx) / (cscx + cotx)
Step 2:
Let's simplify each fraction separately.
For the first fraction (secx + tanx) / (cscx - cotx):
To combine the fractions, we need to find a common denominator, which is (cscx - cotx)(cscx + cotx). Thus, we get:
(secx + tanx)(cscx + cotx) / [(cscx - cotx)(cscx + cotx)]
Now, let's simplify the second fraction (secx - tanx) / (cscx + cotx) in the same manner:
(secx - tanx)(cscx - cotx) / [(cscx + cotx)(cscx - cotx)]
Step 3:
Now, let's combine the two fractions by subtracting them:
[(secx + tanx)(cscx + cotx) - (secx - tanx)(cscx - cotx)] / [(cscx - cotx)(cscx + cotx)]
Step 4:
Expanding this expression further, we get:
[secxcscx + secxcotx + tanxcscx + tanxcotx - secxcscx + secxcotx - tanxcscx + tanxcotx] / [(cscx - cotx)(cscx + cotx)]
Step 5:
Simplifying the terms, we can cancel out some terms:
[2secxcotx] / [(cscx - cotx)(cscx + cotx)]
Step 6:
Using the identities secx = 1/cosx and cotx = cosx/sinx, we can simplify this further:
[2(1/cosx)(cosx/sinx)] / [(cscx - cotx)(cscx + cotx)]
Step 7:
Further simplifying the expression, we have:
[2/sinx] / [(cscx - cotx)(cscx + cotx)]
Step 8:
Using the identity cscx = 1/sinx, we can simplify once more:
2 / [(cscx - cotx)(cscx + cotx)] = 2(secx + cscx)
Therefore, we have manipulated the left side to be equal to the right side:
(secx + tanx) / (cscx - cotx) - (secx - tanx) / (cscx + cotx) = 2(secx + cscx)
I hope this step-by-step explanation helps you understand how to solve proving identities problems. If you have any more questions, feel free to ask!