1. Simplify the expression.
[csc^2(x-1)]/[1+sin x]
a. csc x+1
b. csc x(csc x-1)
c. sin^2 x-csc x****
d. csc^2 x-cos xtan x
2. Which of the following expressions can be used to complete the equation below?
sec x/1+cot^2 x
a. tan x
b. tan^2 x
c. tan x cos x
d. tan x sin x****
3. Simplify the following expression.
cot^2 x sec x-cos x
a. cos x cot^2 x
b. cos x csc^2 x
c. sin x-sec x
d. csc^2 x(1-cos x)****
Damon can you help check please?
Did you ever get the answer>
1. To simplify the expression [csc^2(x-1)]/[1+sin x], we need to use the trigonometric identities. Here's how:
First, let's simplify the numerator, csc^2(x-1). Using the reciprocal identity, csc^2(x) is equal to 1/sin^2(x). So, we can rewrite the numerator as 1/sin^2(x-1).
Next, let's simplify the denominator, 1+sin x.
Now, we can rewrite the expression as (1/sin^2(x-1))/(1+sin x).
To simplify further, we can use the division of fractions rule. When dividing by a fraction, we can multiply by the reciprocal of that fraction. So, we can rewrite the expression as (1/sin^2(x-1)) * (1/(1+sin x)).
Expanding further, the expression simplifies to 1/(sin^2(x-1) * (1+sin x)).
At this point, we don't see any other simplifications we can make, so the answer is option c. sin^2 x - csc x.
2. To find the expression that can complete the equation sec x/1+cot^2 x, we need to manipulate the given equation using trigonometric identities. Here's how:
Looking at the term in the denominator, cot^2 x, we can rewrite it as 1/tan^2 x using the reciprocal identity.
The equation becomes sec x / (1 + 1/tan^2 x).
To simplify further, we need to add the fractions. To add fractions with different denominators, we must first find a common denominator. In this case, the common denominator is tan^2 x.
So, the equation becomes (sec x * tan^2 x + 1) / tan^2 x.
Expanding further, we get (sin x / cos x) * tan^2 x + 1 / tan^2 x.
Using the identity tan x = sin x / cos x, the equation simplifies to sin x * tan x + 1 / tan^2 x.
This matches with option d. tan x sin x.
3. To simplify the expression cot^2 x sec x - cos x, we need to manipulate the expression using trigonometric identities. Here's how:
First, let's simplify the term cot^2 x sec x. Using the identity cot x = cos x / sin x and sec x = 1 / cos x, we can rewrite the expression as (cos x / sin x)^2 * (1 / cos x).
Simplifying further, (cos^2 x / sin^2 x) * (1 / cos x) = cos x / sin x.
Now, let's substitute this back into the original expression. The expression becomes cos x / sin x - cos x.
To simplify further, we can combine the terms over a common denominator. The expression becomes (cos x - cos x * sin x) / sin x.
Factoring out cos x from the numerator, we get cos x(1 - sin x) / sin x.
This matches with option d. csc^2 x(1 - cos x).
#1 nope - how did you try it?
#2 ok
#3 nope