cot²x (sec²x - 1)

Question

LHS: cot²x (sec²x - 1)

= cot²x (tan²x) (find the rule)

= cos²x/sin²x ⋅ sin²x/cos²x (find the rule)

= 1 (find the rule)

(Algebra, Reciprocal, Quotient, Pythagorean, or Odd/Even)

Answer 1

Hello Vincent,

For your question breaking it down line for line.

= cot²x (tan²x) (find the rule): Recall the pythragorean identity cos²(x) + sin²(x) = 1 and you're trying to get a sec²(x) from it since you start off with sec²x - 1. Now recall that 1/cos²(x) = sec²(x) and you can achieve that by dividing both sides of your pythragorean identity with cos²(x). Doing so will yield: 1 + tan²(x) = sec²(x). Using a little bit algebra (moving the 1 to the RHS of your modified pythragorean identity,

you will get: tan²(x) = sec²(x) - 1 and you can see now how the substitution was made.

= cos²x/sin²x ⋅ sin²x/cos²x (find the rule): Now recall that cot(x) = cos(x)/sin(x) and if you square both sides, you will see how cot²(x) was changed to cos²(x)/sin²(x). It's going to be a similar story with the tan²(x) that was substituted out with sin²(x)/cos²(x).

= 1 (find the rule): Multiplying the above is just going to yield 1. I would say the rule is just multiplying by a very specific reciprocal.

I will leave you to piece it together.

Hope this helps!