Question

Sin^2 a (1- ctg^2a)=

Answer 1

$Sin^{2}\alpha*(1-Ctg^{2}\alpha)=Sin^{2}\alpha*(1-\frac{Cos^{2}\alpha}{Sin^{2}\alpha})=Sin^{2}\alpha*\frac{Sin^{2}\alpha-Cos^{2}\alpha}{Sin^{2}\alpha} =\\\\=Sin^{2}\alpha-Cos^{2}\alpha =-\underbrace{(Cos^{2}\alpha-Sin^{2}\alpha)}_{Cos2\alpha}=\boxed{-Cos2\alpha }$

Answer 2

To simplify the expression sin^2(a)(1 - ctg^2(a)), we can use trigonometric identities.

1. First, let's express the cotangent (ctg) in terms of sine and cosine: ctg(a) = cos(a)/sin(a)

2. Now, we'll use the Pythagorean identity for sine and cosine: sin^2(a) + cos^2(a) = 1

3. We can isolate cos^2(a) from this identity: cos^2(a) = 1 - sin^2(a)

4. Substitute this expression for cos^2(a) into the expression: sin^2(a)(1 - (cos(a)/sin(a))^2)

5. Now, expand the square of the cotangent and simplify: sin^2(a)(1 - cos^2(a)/sin^2(a))

6. Substitute the expression for cos^2(a) from step 3: sin^2(a)(1 - (1 - sin^2(a))/sin^2(a))

7. Distribute sin^2(a) into the parentheses: sin^2(a) - (1 - sin^2(a))

8. Distribute the negative sign in front of the second term: sin^2(a) - 1 + sin^2(a)

9. Combine like terms: 2sin^2(a) - 1

So, sin^2(a)(1 - ctg^2(a)) simplifies to 2sin^2(a) - 1.