Question

(1-cos^2a)*(1+ctg^2a)=1

Answer 1

$(1-cos^2a)(1+ctg^2a)=sin^2a\cdot \frac{1}{sin^2a}=1$

Answer 2

The given trigonometric identity is true. Let's prove it step by step:

Starting with the left-hand side: (1 - cos^2(a)) * (1 + ctg^2(a))

Using the trigonometric identities: cos^2(a) + sin^2(a) = 1 (1) 1 + ctg^2(a) = csc^2(a) (2)

Substitute (2) into the expression: (1 - cos^2(a)) * csc^2(a)

Using the identity: 1 - cos^2(a) = sin^2(a) (3)

Substitute (3) into the expression: sin^2(a) * csc^2(a)

Now, using the identity: csc(a) = 1 / sin(a) (4)

Substitute (4) into the expression: sin^2(a) * (1 / sin(a))^2

Simplify: sin^2(a) * 1 / sin^2(a)

The sin^2(a) terms cancel out: 1

Therefore, the left-hand side simplifies to 1, which matches the right-hand side. This confirms that the given trigonometric identity (1 - cos^2(a)) * (1 + ctg^2(a)) = 1 is indeed true.