Question

Sin^2(45+a)-sin^2(30-a)-sin 15-cos(15+2a)=sin2a

Answer 1

........... ..........,,.

Answer 2

Let's solve the trigonometric expression step by step:

1. First, let's simplify the left-hand side of the equation: sin^2(45 + a) - sin^2(30 - a) - sin 15 - cos(15 + 2a)

2. Use trigonometric identities to simplify further:

   a) sin^2(x) - cos^2(x) = 1 for any angle x. b) sin(45) = cos(45) = 1/sqrt(2) c) sin(30) = 1/2 d) cos(30) = sqrt(3)/2 e) sin(15) = (sqrt(6) - sqrt(2)) / 4 f) cos(15) = (sqrt(6) + sqrt(2)) / 4

3. Applying the above identities, the expression becomes:

   (1/2)^2 - (sqrt(3)/2)^2 - [(sqrt(6) - sqrt(2)) / 4] - [(sqrt(6) + sqrt(2)) / 4] = 1/4 - 3/4 - sqrt(6)/4 + sqrt(2)/4 - sqrt(6)/4 - sqrt(2)/4

   The sqrt(6) and sqrt(2) terms cancel each other out.

   = 1/4 - 3/4 = -2/4 = -1/2

4. Now, let's simplify the right-hand side of the equation: sin(2a)

5. Use the double angle identity for sine: sin(2a) = 2 * sin(a) * cos(a)

6. From the previous trigonometric identities, we know: sin(15) = (sqrt(6) - sqrt(2)) / 4 cos(15) = (sqrt(6) + sqrt(2)) / 4

7. Substituting these values into the double angle identity: sin(2a) = 2 * [sin(a) * cos(a)] = 2 * [(sqrt(6) - sqrt(2)) / 4 * (sqrt(6) + sqrt(2)) / 4]

   Using the difference of squares (a^2 - b^2 = (a + b)(a - b)): sin(2a) = 2 * [(6 - 2) / 16] = 2 * (4 / 16) = 2 * (1 / 4) = 1 / 2

Now we have the left-hand side as -1/2 and the right-hand side as 1/2.

Since they are not equal, the given equation is incorrect. There might be an error in the original expression or a mistake in the equation you provided. Double-check the expression and ensure that all the trigonometric functions and their arguments are correct.