Question

Cos4a + 1 = 1/2 sin4a * (ctga - tga)

Answer 1

Ответ

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

Объяснение:

1-2sin^2(2a) +1 = 1/2 sin4a* (2/ tg2a)

2*(1-sin^2(2a)) =sin 4a /tg2a

2*(1-sin^2(2a)) = (2*sin 2a* cos2a) / (sin 2a/cos 2a)

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

Answer 2

To solve the equation cos(4a) + 1 = (1/2)sin(4a)(cot(a) - tan(a)), we can simplify both sides of the equation and then attempt to find the values of 'a' that satisfy the equation.

Let's start by simplifying the right-hand side (RHS) of the equation:

RHS = (1/2)sin(4a)(cot(a) - tan(a)) = (1/2)(sin(4a)/cos(a))(cos(a)/sin(a) - sin(a)/cos(a)) = (1/2)(sin(4a) - sin^2(a))/sin(a)cos(a)

Now, let's simplify the left-hand side (LHS) of the equation:

LHS = cos(4a) + 1

Next, we'll use the double-angle formulas to express cos(4a) and sin(4a) in terms of cos(a) and sin(a):

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

Substituting these expressions into the equation:

LHS = 1 - 2sin^2(2a) + 1 = 2 - 2sin^2(2a)

Now, we can rewrite the equation as:

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

Multiplying both sides by 2sin(a)cos(a) to eliminate the denominators:

4sin(a)cos(a) - 4sin^3(a)cos(a) = sin(4a) - sin^2(a)

Using the double-angle formulas again:

4sin(a)cos(a) - 4sin^3(a)cos(a) = 2sin(2a)cos(2a) - sin^2(a)

Expanding sin^2(a) and rearranging the terms:

4sin(a)cos(a) - 4sin^3(a)cos(a) = 2sin(2a)cos(2a) - (1 - cos^2(a))

Using the identity sin(2a) = 2sin(a)cos(a):

4sin(a)cos(a) - 4sin^3(a)cos(a) = 2(2sin(a)cos(a))(cos^2(a) - sin^2(a)) - (1 - cos^2(a))

Simplifying further:

4sin(a)cos(a) - 4sin^3(a)cos(a) = 4sin(a)cos(a)(cos^2(a) - sin^2(a)) - 1 + cos^2(a)

Now, we can cancel out the common factors of sin(a)cos(a) from both sides:

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

Expanding and rearranging:

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

Simplifying further:

4 = 3cos^2(a)

Dividing both sides by 3:

4/3 = cos^2(a)

Taking the square root of both sides:

±√(4/3) = cos(a)

Since cos(a) cannot be greater than 1 or less than -1, we only consider the positive square root:

cos(a) = √(4/3)

To find the values of 'a', we can take the inverse cosine (cos^(-1)) of both sides:

a = cos^(-1)(√(4/3))

Please note that the value of 'a' can take multiple angles within the range of [0, 2π) that satisfy the equation.