Question 814827
Given;
(1) {{{1/(1-cosx) + 1/(1+cosx) = 2csc^2(x)}}}
First combine the fractions on the left side of (1) by cross multiplying to get
(2) {{{((1+cosx) + (1-cosx))/((1-cosx)*(1+cosx)) = 2csc^2(x)}}}
Now add the numerator terms and multiply the denominator factors of the left side of (2) to get
(3) {{{(1+1+cosx-cosx)/(1+cosx-cosx-cos^2(x)) = 2csc^2(x)}}} or
(4){{{(2)/(1-cos^2(x)) = 2csc^2(x)}}}
Now use the identity
(5) {{{sin^2(x) + cos^2(x) = 1}}} to get
(6) {{{sin^2(x) = 1 - cos^2(x)}}}
Now substitute (6) into (4) to get
(7){{{(2)/(sin^2(x)) = 2csc^2(x)}}}
Now use the reciprocal identity
(8){{{1/(sin(x)) = csc(x)}}} to get
(9){{{1/(sin^2(x)) = csc^2(x)}}} or
by multiplying both sides of (9) by 2 we get
(10) {{{2/(sin^2(x)) = 2csc^2(x)}}}
Now substitute (10) into (7) to get
(11) {{{2csc^2(x) = 2csc^2(x)}}} Q.E.D.