Question

1) 1-1/sin^2L 2)4-tgL*ctgL 3)cos^2b-cos^2bsin^2b

Answer 1

$1-\frac{1}{sin^2a}=\frac{sin^2a}{sin^2a}-\frac{1}{sin^2a}=\frac{sin^2a-1}{sin62a}=\frac{-(1-sin^2a)}{sin^2a}=\frac{-cos^2a}{sin^2a}=-ctg^2a\\\\4-tga*ctga=4-\frac{sina}{cosa}*\frac{cosa}{sina}= 4-1=3\\\\cos^2\beta-cos^2\beta*sin^2\beta=cos^2\beta(1-sin^2\beta)=cos^2\beta*cos^2\beta=cos^4\beta$

Answer 2

Let's simplify each of the given expressions:

1. 1 - 1/sin^2(L): To simplify this expression, we need to use the trigonometric identity: 1 - sin^(-2)(x) = cos^2(x)

So, for our expression: 1 - 1/sin^2(L) = cos^2(L)

2. 4 - tg(L) * ctg(L): To simplify this expression, we use the trigonometric identity: tan(x) * cot(x) = 1

So, for our expression: 4 - tg(L) * ctg(L) = 4 - 1 = 3

3. cos^2(b) - cos^2(b) * sin^2(b): To simplify this expression, we use the trigonometric identity: cos^2(x) - cos^2(x) * sin^2(x) = cos^2(x) * (1 - sin^2(x))

Again, we use the trigonometric identity: 1 - sin^2(x) = cos^2(x)

So, for our expression: cos^2(b) - cos^2(b) * sin^2(b) = cos^2(b) * (1 - sin^2(b)) = cos^2(b) * cos^2(b) = cos^4(b)

To summarize:

1. 1 - 1/sin^2(L) simplifies to cos^2(L).
2. 4 - tg(L) * ctg(L) simplifies to 3.
3. cos^2(b) - cos^2(b) * sin^2(b) simplifies to cos^4(b).