Question

Тригонометрия Доказать тождество cos^2a/1-cos^2a = ctg^2a (^2– это квадрат; а- альфа; / - деление)

Answer 1

Cos² α / (1-cos² α) = cos² α / sin² α = ctg² α

ctg² α = ctg² α

что и требовалось доказать

Answer 2

Trigonometric Identity Proof: cos^2a / (1 - cos^2a) = ctg^2a

To prove the trigonometric identity cos^2a / (1 - cos^2a) = ctg^2a, we can use the following trigonometric identities:

1. cos^2a = 1 - sin^2a 2. ctg^2a = 1 / tan^2a

Using these identities, we can manipulate the given expression to match the form of the second identity.

Starting with cos^2a / (1 - cos^2a), we can substitute 1 - sin^2a for cos^2a:

cos^2a / (1 - cos^2a) = (1 - sin^2a) / (1 - (1 - sin^2a))

Simplifying the denominator gives:

cos^2a / (1 - cos^2a) = (1 - sin^2a) / sin^2a

Now, we can use the identity ctg^2a = 1 / tan^2a to rewrite the expression:

cos^2a / (1 - cos^2a) = (1 - sin^2a) / sin^2a = 1 / (sin^2a) - 1 = ctg^2a

Therefore, the trigonometric identity cos^2a / (1 - cos^2a) = ctg^2a is proven.