Can anyone please explain these problems to me?? I really need help!I need to simplify these trig expressions but I don't understand how my teacher is doing it at all and I have a quiz on this tomorrow! I've really been trying but I just don't get it!
1)Mulitply:
[cosx/(1-cosx)][1+sinx/(1+sinx)]
2)Rewrite over common denominator:
[1/(1-cosx)]+[1/(1+cosx)]
3)Establish the identity:
[(1-cot^2x)/(1+cot^2x)]+1=2sin^2x
cos/(1-cos) * ((1+sin)+sin)/(1+sin)
cos(1+2sin)/((1-cos)(1+sin))
Not sure how to make it any simpler. No chance of a typo in the problem?
----------------------------
((1+cos) + (1-cos))/(1-cos)(1+cos)
2/sin^2
----------------------
(1-cos^2/sin^2)/(1+cos^2/sin^2) + 1
(sin^2 - cos^2)/sin^2) / ((sin^2+cos^2)/sin^2) + 1
(sin^2-cos^2)/(sin^2+cos^2) + 1
(sin^2 - cos^2) + 1
sin^2 + 1-cos^2
sin^2+sin^2
2sin^2
Of course, I'd be happy to explain these problems to you! Understanding trigonometric expressions and identities can be challenging, but with some practice and guidance, you'll be able to simplify them with ease.
1) Multiply: [cosx/(1-cosx)][1+sinx/(1+sinx)]
To solve this problem, we'll use the concept of multiplying fractions. Here's how you can do it step by step:
Step 1: Start by simplifying each fraction individually.
For the first fraction, cosx/(1-cosx), there is no immediate simplification possible.
For the second fraction, sinx/(1+sinx), we can simplify it by multiplying both the numerator and denominator by (1-sinx), which is the conjugate of (1+sinx). This gives us:
sinx * (1-sinx) / [(1+sinx)(1-sinx)]
Simplifying the numerator further, we get:
sinx - sin^2x / [(1+sinx)(1-sinx)]
Step 2: Now that we have simplified fractions, we can multiply them together.
Multiply the numerators together: cosx * (sinx - sin^2x).
Multiply the denominators together: (1-cosx)(1+sinx)(1-sinx).
Step 3: Simplify the result obtained in step 2.
To further simplify the numerator, we can distribute cosx into each term:
cosx * sinx - cosx * sin^2x
Now, we arrange the terms in descending order:
-cosx * sin^2x + cosx * sinx
Combining the terms, we have:
cosx * sinx - cosx * sin^2x
Step 4: Simplify the denominator if possible.
Multiply out the denominator:
(1 - cos^2x)(1 - sin^2x)
Using the Pythagorean identities sin^2x + cos^2x = 1, we can further simplify the denominator:
(1 - cos^2x)(1 - sin^2x) = sin^2x * cos^2x
Step 5: Combining the numerator and denominator, we get the final answer:
(cosx * sinx - cosx * sin^2x) / (sin^2x * cos^2x)
This is the simplified expression.
Now, let's move on to the next problem.
2) Rewrite over a common denominator: [1/(1-cosx)] + [1/(1+cosx)]
To solve this problem, we'll follow these steps:
Step 1: Find a common denominator for the two fractions, which in this case is (1 - cosx)(1 + cosx).
Step 2: Rewrite each fraction with the common denominator.
For the first fraction, multiply the numerator and denominator by (1 + cosx):
[1 * (1 + cosx)] / [(1 - cosx)(1 + cosx)]
For the second fraction, multiply the numerator and denominator by (1 - cosx):
[1 * (1 - cosx)] / [(1 - cosx)(1 + cosx)]
Step 3: Simplify the numerators:
[(1 + cosx) + (1 - cosx)] / [(1 - cosx)(1 + cosx)]
The numerators simplify to 2.
Step 4: Simplify the denominator:
[(1 - cosx)(1 + cosx)] = (1 - cos^2x)
Using the Pythagorean identity sin^2x + cos^2x = 1, we can rewrite cos^2x as 1 - sin^2x:
(1 - cos^2x) = (1 - (1 - sin^2x)) = sin^2x
Step 5: Combine the simplified numerator and denominator:
2 / sin^2x
This is the simplified expression.
Now, let's move on to the last problem.
3) Establish the identity: [(1 - cot^2x)/(1 + cot^2x)] + 1 = 2sin^2x
To prove this identity, we need to simplify both sides and show that they are equal. Here's the step-by-step process:
Left-hand side:
[(1 - cot^2x)/(1 + cot^2x)] + 1
Step 1: Simplify the numerator and denominator of the first fraction:
[(1 - (cos^2x/sin^2x))/(1 + (cos^2x/sin^2x))] + 1
Step 2: Combine the fractions.
[(sin^2x - cos^2x)/(sin^2x + cos^2x)] + 1
Using the Pythagorean identity sin^2x + cos^2x = 1, we can simplify further:
[sin^2x - cos^2x + sin^2x + cos^2x] / (sin^2x + cos^2x)
Step 3: Combine like terms:
[2sin^2x] / (sin^2x + cos^2x)
Using the Pythagorean identity sin^2x + cos^2x = 1, we simplify the denominator to 1:
[2sin^2x] / 1
The left-hand side simplifies to 2sin^2x.
Now, let's simplify the right-hand side:
Right-hand side:
2sin^2x
Both sides are equal, and therefore, the identity [(1 - cot^2x)/(1 + cot^2x)] + 1 = 2sin^2x holds true.
I hope this explanation helps you understand how to solve these trigonometric expressions. If you have any further questions, feel free to ask! Good luck with your quiz!