Question

Cos18*cos36 + (sin36)^2/2cos18

Answer 1

Ответ:

Пошаговое объяснение:

cos18*(cos²18-sin²18)+(4sin²18*cos²18)/2cos18=cos18*(cos²18-sin²18)+2sin²18cos18=cos18(cos²18-sin²18+2sin²18)=cos18(cos²18+sin²18)=cos18

Answer 2

To simplify the given expression, we can use trigonometric identities to rewrite the terms in a more manageable form.

Let's start by recalling some trigonometric identities:

1. cos(2θ) = 2cos^2(θ) - 1
2. sin^2(θ) = 1 - cos^2(θ)
3. cos(90° - θ) = sin(θ)
4. sin(2θ) = 2sin(θ)cos(θ)

Now, let's simplify the expression step by step:

1. cos(18°)cos(36°): Using the double-angle formula (cos(2θ)), we can rewrite cos(36°) as cos(218°): cos(36°) = cos(2*18°) = 2cos^2(18°) - 1

So, the first term becomes: cos(18°)cos(36°) = cos(18°)(2cos^2(18°) - 1)

2. (sin(36°))^2: Using the identity sin^2(θ) = 1 - cos^2(θ), we can find sin(36°)^2: (sin(36°))^2 = 1 - cos^2(36°)

3. 2*cos(18°): Using the identity cos(90° - θ) = sin(θ), we can rewrite cos(18°) as sin(72°): cos(18°) = sin(90° - 18°) = sin(72°)

4. (sin(36°))^2/2cos(18°): Now, we can substitute the values from steps 2 and 3 into the expression: (sin(36°))^2/2cos(18°) = (1 - cos^2(36°))/2*sin(72°)

5. Final expression: Substitute the value of cos(36°) from step 1 into the expression from step 4: (cos(18°)(2cos^2(18°) - 1) + (1 - cos^2(36°))/2sin(72°)

Now, we have an expression entirely in terms of cos(18°). Let's simplify it further:

Let's define x = cos(18°):

Expression = (x*(2x^2 - 1) + (1 - (2x^2 - 1))/2*sin(72°)

Expression = (2x^3 - x) + (2 - 2x^2)/2*sin(72°)

Expression = 2x^3 - x + (2 - 2x^2)/2*sin(72°)

Expression = 2x^3 - x + (1 - x^2)*sin(72°)

The expression 2x^3 - x + (1 - x^2)*sin(72°) is the simplified form of the given expression.