Question

2sin(Пи/4+a)*sin(Пи/4-a)+sin^2a Тригонометрия. Помогите Please

Answer 1

(cos^2(x)-sin^2(x))*(cos^2(x)+sin^2(x))+sin2x=cos2x+sin2x левая часть тождества №2*(cos2a*№2/2 +sin2x*№2/2)=cos2x+sin2x правая часть тождества 

Answer 2

Trigonometry: Simplifying the Expression 2sin(π/4+a)sin(π/4-a)+sin^2a

To simplify the expression 2sin(π/4+a)sin(π/4-a)+sin^2a, we can use trigonometric identities and properties. Let's break it down step by step:

Step 1: Expand the product of the two sine terms using the identity sin(A+B) = sinAcosB + cosAsinB.

The expression becomes: 2(sin(π/4)cos(a) + cos(π/4)sin(a)) * (sin(π/4)cos(-a) + cos(π/4)sin(-a)) + sin^2(a)

Step 2: Simplify the trigonometric values of π/4.

The values of sin(π/4) and cos(π/4) are both equal to 1/√2.

The expression becomes: 2(1/√2 * cos(a) + 1/√2 * sin(a)) * (1/√2 * cos(-a) + 1/√2 * sin(-a)) + sin^2(a)

Step 3: Simplify the cosine and sine of negative angles.

cos(-a) = cos(a) and sin(-a) = -sin(a).

The expression becomes: 2(1/√2 * cos(a) + 1/√2 * sin(a)) * (1/√2 * cos(a) - 1/√2 * sin(a)) + sin^2(a)

Step 4: Simplify the product of the two terms.

Using the identity (a+b)(a-b) = a^2 - b^2, we can simplify the product of the two terms.

The expression becomes: 2(1/√2 * cos(a))^2 - (1/√2 * sin(a))^2 + sin^2(a)

Step 5: Simplify the squares.

(1/√2)^2 = 1/2, and sin^2(a) is already simplified.

The expression becomes: 2(1/2 * cos^2(a)) - (1/2 * sin^2(a)) + sin^2(a)

Step 6: Combine like terms.

The expression becomes: cos^2(a) - 1/2 * sin^2(a) + sin^2(a)

Step 7: Simplify further.

cos^2(a) - 1/2 * sin^2(a) + sin^2(a) simplifies to: cos^2(a) + 1/2 * sin^2(a)

So, the simplified expression is cos^2(a) + 1/2 * sin^2(a).

Please note that the simplified expression may vary depending on the context and the specific trigonometric identities used.