Question

Найти ответ cos²74-cos²30+sin²74​

Answer 1

Ответ:

cos^2 45°+sin^2 74°+cos^2 74°=

cos^2 45°+1=1/2+1=3/2

Объяснение:

Answer 2

To find the value of the expression cos²74 - cos²30 + sin²74, we can use trigonometric identities to simplify it.

1. Using the Pythagorean identity: sin²θ + cos²θ = 1

We know that sin²74 + cos²74 = 1.

2. Using the angle addition formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

We can rewrite cos²74 as cos²(30 + 44) using the fact that 74 = 30 + 44.

cos²(30 + 44) = cos²30cos²44 - sin²30sin²44

Since cos²30 = (sqrt(3)/2)² = 3/4 and sin²30 = (1/2)² = 1/4, we can substitute these values:

cos²(30 + 44) = (3/4) * cos²44 - (1/4) * sin²44

Now, we need to find the values of cos²44 and sin²44.

Using the double-angle formula: cos(2θ) = 2cos²θ - 1

cos(2 * 22) = 2 * cos²22 - 1

cos(44) = 2 * cos²22 - 1

Now, to find cos²22, we can use the half-angle formula: cos(θ/2) = ± sqrt((1 + cosθ)/2)

Since 0° < 22° < 90°, cos(θ/2) is positive. Therefore,

cos22 = sqrt((1 + cos44)/2)

Now we can find cos²22:

cos²22 = (1 + cos44)/2

Substitute the value of cos44:

cos²22 = (1 + [2 * cos²22 - 1])/2

Simplify:

2 * cos²22 = 1 + 2 * cos²22 - 1

cos²22 = 1/2

Now, substitute the value of cos²22 and sin²30 into cos²(30 + 44):

cos²(30 + 44) = (3/4) * (1/2) - (1/4) * sin²44

cos²(30 + 44) = 3/8 - (1/4) * sin²44

Now, we need to find sin²44. Using the Pythagorean identity: sin²θ + cos²θ = 1

sin²44 + cos²44 = 1

Since we already found cos²44 to be 1/2:

sin²44 + 1/2 = 1

sin²44 = 1 - 1/2

sin²44 = 1/2

Now, substitute the value of sin²44 back into the expression:

cos²(30 + 44) = 3/8 - (1/4) * (1/2)

cos²(30 + 44) = 3/8 - 1/8

cos²(30 + 44) = 2/8

cos²(30 + 44) = 1/4

Finally, we can find the value of the original expression:

cos²74 - cos²30 + sin²74 = 1 - 3/4 + 1/4 = 1 - 3/4 + 1/4 = 1.

The final answer is 1.