Use double-angle or half-angle formula to simplify #(2-csc^2(x))/(csc^2(x))# ?

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Madelyn Adams · Accepted Answer

#(2-csc^2x)/csc^2x=-cos2x# We have #csc^2x=1/sin^2x# Then #(2-csc^2x)/csc^2x# #=(2-1/sin^2x)/csc^2# #=((2sin^2x-1)/sin^2x)/(1/sin^2x)# #=2sin^2x-1# #=-(1-2sin^2x)# #=-cos2x#

Brooklyn Anderson · Answer

undefined To simplify the expression (2 - csc^2(x)) / csc^2(x), we can rewrite csc^2(x) as 1 / sin^2(x). Then, we substitute csc^2(x) with its reciprocal form:

(2 - (1 / sin^2(x))) / (1 / sin^2(x))

To get rid of the complex fraction, we multiply the numerator and denominator by sin^2(x):

(2sin^2(x) - 1) / 1

Now, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) with 1 - cos^2(x):

(2(1 - cos^2(x)) - 1) / 1

Expanding and simplifying:

(2 - 2cos^2(x) - 1) / 1

(1 - 2cos^2(x)) / 1

Now, we can apply the double-angle formula for cosine: cos(2x) = 1 - 2sin^2(x).

Substitute cos^2(x) with (1 - sin^2(x)):

1 - 2(1 - sin^2(x))

1 - 2 + 2sin^2(x)

2sin^2(x) - 1

Therefore, the simplified expression is 2sin^2(x) - 1.