Question

Cos^4a-sin^4a=1-2sin^2a

Answer 1

(cosa)^4 - (sina)^4 = ((cosa)^2 - (sina)^2) * ((cosa)^2 + (sina)^2) (разность квадратов)

= cos(2a) * 1 = 1 - 2*(sina)^2

второе не очень понятно: sin18^0 - это sin 18 градусов???

sin18 * sin12 = sin(30-12) * sin12 = (sin30*cos12 - cos30*sin12) * sin12 = (1/2 * cos12 - корень(3)/2 * sin12) * sin12 = 1/2 * cos12 * sin12 - корень(3)/2 * sin12 * sin12 = 1/4 * sin24 - корень(3)/2 * (sin12)^2 = 1/4 * sin24 - корень(3)/4 * (1-cos24) =

1/4 * sin24 + корень(3)/4 * cos24 - корень(3)/4

Answer 2

Simplifying the equation

To prove the equation cos^4(a) - sin^4(a) = 1 - 2sin^2(a), let's start by simplifying both sides of the equation separately.

Simplifying the left-hand side (LHS)

We'll start by simplifying the left-hand side (LHS) of the equation cos^4(a) - sin^4(a).

Recall the identities for the fourth power of cosine and sine:

cos^4(a) = (cos^2(a))^2 sin^4(a) = (sin^2(a))^2

Using these identities, we can rewrite the LHS of the equation as:

cos^4(a) - sin^4(a) = (cos^2(a))^2 - (sin^2(a))^2

Now, let's apply the difference of squares identity:

a^2 - b^2 = (a + b)(a - b)

Using this identity, we can rewrite the LHS as:

cos^4(a) - sin^4(a) = [(cos^2(a) + sin^2(a))(cos^2(a) - sin^2(a))]

Recall the Pythagorean identity:

cos^2(a) + sin^2(a) = 1

Using this identity, we can simplify further:

cos^4(a) - sin^4(a) = [1(cos^2(a) - sin^2(a))]

Recall the difference of squares identity:

a^2 - b^2 = (a + b)(a - b)

We can apply this identity again to get:

cos^4(a) - sin^4(a) = (cos^2(a) + sin^2(a))(cos^2(a) - sin^2(a))

Finally, using the Pythagorean identity, we know that cos^2(a) + sin^2(a) = 1. Substituting this identity into the equation, we get:

cos^4(a) - sin^4(a) = (1)(cos^2(a) - sin^2(a))

Thus, the left-hand side simplifies to:

cos^4(a) - sin^4(a) = cos^2(a) - sin^2(a)

Simplifying the right-hand side (RHS)

Next, let's simplify the right-hand side (RHS) of the equation 1 - 2sin^2(a).

The RHS is already in a simplified form, so we don't need to perform any further simplification.

Comparing the LHS and RHS

Now, let's compare the simplified LHS and RHS:

cos^4(a) - sin^4(a) = cos^2(a) - sin^2(a) = 1 - 2sin^2(a)

As we can see, the left-hand side (LHS) is equal to the right-hand side (RHS), thus proving the equation.

Therefore, we have shown that cos^4(a) - sin^4(a) = 1 - 2sin^2(a) is true for all values of angle 'a'.