cot²x (sec²x - 1) = 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, Odd/Even)

Answer 1

1.

The principal Pythagorean identity:

sin²x + cos² x = 1

Divide by cos² x (quotient):

(sin² x / cos² x) + (cos² x / cos² x ) = ( 1 / cos² x)

tan² x + 1 = sec² x

Subtract 1 from both sides:

tan² x = ( sec² x ) - 1

2.

Algebra and reciprocal.

cot² x = [ (cos x) / (sin x) ]² = (cos² x) / (sin² x)

tan² x = 1 / (cot² x) = (sin² x) / (cos² x)

3.

Using the associative and commutative laws of multiplication (algebra):

(a/b)×(b/a) = a×(b/b)/a = a × 1 / a = a/a = 1

where a≠0 and b≠0. Reciprocal (multiplicative identity) and algebra.

Answer 2

cot²x(sec²x - 1) = cot²xsec²x - cot²x = (cos²x / sin²x)(1 / cos²x) - cos²x / sin²x)

= 1 / sin²x - cos²x / sin²x = (1 - cos²x) / sin²x = sin²x / sin²x = 1