SOLUTION: hi could you please help me to verify this identity (cot^3 x

SOLUTION: hi could you please help me to verify this identity (cot^3 x - tan^3 x)/(sec^2x + cot^2x) = 2cot(2x)

Question 376402: hi could you please help me to verify this identity

(cot^3 x - tan^3 x)/(sec^2x + cot^2x) = 2cot(2x)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

First, we can use the pattern, , to factor the numerator:

Since cot and tan are reciprocals, cot(x)*tan(x) = 1:

Since :

We can now see that the fraction will reduce:

leaving:

Rewriting the left side using sin and cos (a common technique use in these problems) we get:

Since the right side has one term, we probably want one term on the left side, too. So we will add the fractions. Of course we need common denominators first:

Now we can subtract:

The numerator exactly matches the formula for cos(2x). The denominator is close to sin(2x). sin(2x) = 2sin(x)cos(x). If we multiply both sides by 1/2 we get: . Substituting these into our fraction we get:

We can get rid of the fraction in the denominator by multiplying the numerator and denominator by 2:

giving:

The fraction is cot(2x):
2cot(2x) = 2cot(2x) Done!
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