Multiply then use fundamental identities to simplify the expression below and determine which of the following is not equivalent (2 - 2cos x)(2 + 2cos x)
a.4 - 4cos^2 x
b . 4 - cos^2 x
c.4/(1 + cot^2 x)
d. 4sin^2 x
e.4/(csc^2 x)
recall that (a-b)(a+b) = a^2-b^2
so, 2^2 - (2cosx)^2 = 4 - 4cos^2x = 4(1-cos^2x) = 4sin^2x
only B is not equivalent to that
To simplify the expression (2 - 2cos x)(2 + 2cos x), we can use the difference of squares formula, which states that (a - b)(a + b) = a^2 - b^2.
By applying this formula to the expression, we get:
(2 - 2cos x)(2 + 2cos x) = (2)^2 - (2cos x)^2
= 4 - 4cos^2 x
Now, we need to determine which option among a, b, c, d, and e is not equivalent to 4 - 4cos^2 x.
Let's evaluate each option:
a. 4 - 4cos^2 x - This option is equivalent to the simplified expression (2 - 2cos x)(2 + 2cos x).
b. 4 - cos^2 x - This option is not equivalent since it lacks the multiplication by 4. Therefore, this option is not equivalent to (2 - 2cos x)(2 + 2cos x).
c. 4/(1 + cot^2 x) - This option represents a different expression and is not equivalent to (2 - 2cos x)(2 + 2cos x).
d. 4sin^2 x - This option is not equivalent to the simplified expression (2 - 2cos x)(2 + 2cos x) since it involves the sine function instead of the cosine function.
e. 4/(csc^2 x) - This option represents a different expression and is not equivalent to (2 - 2cos x)(2 + 2cos x).
Therefore, the option that is NOT equivalent to (2 - 2cos x)(2 + 2cos x) is b. 4 - cos^2 x.
To simplify the expression (2 - 2cos x)(2 + 2cos x), we can use the formula for the difference of squares. The difference of squares formula states that a^2 - b^2 can be factored as (a + b)(a - b).
In this case, let's consider (2 - 2cos x) as a and (2 + 2cos x) as b. Applying the difference of squares formula, we have:
(2 - 2cos x)(2 + 2cos x) = (2)^2 - (2cos x)^2
Now let's simplify this expression step by step:
(2)^2 - (2cos x)^2 = 4 - 4cos^2 x
So the simplified expression is 4 - 4cos^2 x.
Now let's check which option among a, b, c, d, and e is not equivalent to 4 - 4cos^2 x:
a. 4 - 4cos^2 x (This is equivalent to the simplified expression)
b. 4 - cos^2 x (This is different because the coefficient of cos^2 x is 1 and not 4)
c. 4/(1 + cot^2 x) (This is a trigonometric identity, specifically the Pythagorean Identity, and is different from the simplified expression)
d. 4sin^2 x (This is different because it involves sin^2 x instead of cos^2 x)
e. 4/(csc^2 x) (This is a reciprocal identity and is different from the simplified expression)
Therefore, option b, 4 - cos^2 x, is not equivalent to (2 - 2cos x)(2 + 2cos x).