• If asin15+bcos15=square root 2 sin15.cos15 find the values of a and b

We are given the equation:

a sin 15° + b cos 15° = √2 sin 15° cos 15°

Step 1: Express sin 15° cos 15° in a simpler form
Using the identity:

sin A cos B = 1/2 [sin (A + B) + sin (A - B)]

For A = B = 15°:

sin 15° cos 15° = 1/2 [sin (30°) + sin (0°)]
= 1/2 [1/2 + 0]
= 1/2 × 1/2
= 1/4

Thus, our equation becomes:

a sin 15° + b cos 15° = √2 × 1/4
a sin 15° + b cos 15° = √2 / 4

Step 2: Compare Coefficients
The given equation must hold for all values of 15°. This means we can compare the coefficients of sin 15° and cos 15° on both sides.

Since there is no sin 15° term on the right-hand side, its coefficient must be 0:
a = 0

For cos 15°:
b = √2 / 4

Step 3: Conclusion
Thus, the values of a and b are:

a = 0
b = √2 / 4