45) Prove the following identities. Use a separate piece of paper.
b) (csc x − cot x)^2 =1−cos x/1+cos x
To prove the identity (csc x − cot x)^2 =1−cos x/1+cos x, we will start from the left side and simplify it until it matches the right side.
Starting with the left side:
(csc x − cot x)^2
Expanding the square:
(csc x − cot x)(csc x − cot x)
Using the distributive property:
csc x * csc x - cot x * csc x - csc x * cot x + cot x * cot x
Using the reciprocal identities:
(1/sin x) * (1/sin x) - (cos x/sin x) * (1/sin x) - (1/sin x) * (cos x/sin x) + (cos x/sin x) * (cos x/sin x)
Simplifying the fractions:
1/(sin x * sin x) - cos x/(sin x * sin x) - cos x/(sin x * sin x) + cos^2 x/(sin x * sin x)
Combining like terms:
(1 - 2cos x + cos^2 x)/(sin x * sin x)
Using the Pythagorean identity sin^2 x + cos^2 x = 1:
(1 - 2cos x + cos^2 x)/(1 - cos^2 x)
Factoring the numerator as a binomial squared:
((1 - cos x)^2)/(1 - cos^2 x)
Using the difference of squares:
((1 - cos x)^2)/((1 - cos x)(1 + cos x))
Canceling out common factors in the numerator and denominator:
(1 - cos x)/(1 + cos x)
Which is equal to the right side of the identity.
Therefore, we have proven that (csc x − cot x)^2 =1−cos x/1+cos x.