Question

2sin^2 (45 -a) = sin2a

Answer 1

Ответ:

$2 { \sin}^{2} (45 - \alpha ) = 2 {( \sin(45 - \alpha )) }^{2} = \\ = 2 {( \sin(45) \cos( \alpha ) - \sin( \alpha ) \cos(45) )}^{2} = \\ = 2 {( \frac{ \sqrt{2} }{2} \cos( \alpha ) - \frac{ \sqrt{2} }{2} \sin( \alpha ) )}^{2} = \\ = 2 {( \frac{ \sqrt{2} }{2}( \cos( \alpha ) - \sin( \alpha )) }^{2} = \\ = 2 \times \frac{1}{2} {( \cos( \alpha ) - \sin( \alpha )) }^{2} = \\ = { \cos }^{2} \alpha - 2 \sin( \alpha ) \cos( \alpha ) + { \sin}^{2} \alpha = \\ = 1 - \sin(2 \alpha )$

Answer 2

To solve the equation 2sin^2(45 - a) = sin(2a), you can use trigonometric identities and simplifications. Here's how you can do it step by step:

1. Start with the equation: 2sin^2(45 - a) = sin(2a)

2. Apply the double angle identity for sine: sin(2a) = 2sin(a)cos(a)

3. Replace sin(2a) with its equivalent expression: 2sin^2(45 - a) = 2sin(a)cos(a)

4. Divide both sides of the equation by 2 to simplify: sin^2(45 - a) = sin(a)cos(a)

5. Now, let's use the angle sum identity for sine, which states that sin(A - B) = sin(A)cos(B) - cos(A)sin(B). In this case, A = 45 and B = a: sin(45 - a) = sin(45)cos(a) - cos(45)sin(a)

6. Simplify further: (1/√2)(cos(a) - sin(a)) = (1/√2)(cos(a) - sin(a))

7. Divide both sides of the equation by (1/√2): cos(a) - sin(a) = cos(a) - sin(a)

8. Subtract cos(a) from both sides: -sin(a) = -sin(a)

9. Finally, divide both sides by -1 to isolate sin(a): sin(a) = sin(a)

So, the equation sin(a) = sin(a) is true for all values of a. Therefore, the solution to the original equation 2sin^2(45 - a) = sin(2a) is that it holds true for all values of 'a.'