Answer
Explanation:
To prove that sin 15° = (√6-√2)/4, we can use the formula for the sine of the sum of two angles:
sin (A + B) = sin A cos B + cos A sin B
We can set A = 15° and B = 75°, since 75° is the supplement of 15° (that is, 180° - 15°). Then we have:
sin 90° = sin 15° cos 75° + cos 15° sin 75°
Since sin 90° = 1 and cos 75° = (√6-√2)/4, we can simplify the equation to:
1 = sin 15° * (√6-√2)/4 + cos 15° * sin 75°
Since sin 75° = (√6+√2)/4, we can substitute that value in and simplify further:
1 = sin 15° * (√6-√2)/4 + cos 15° * (√6+√2)/4
Combining like terms, we get:
1 = (√6-√2 + √6+√2)/4
Which simplifies to:
1 = (2√6)/4
Which simplifies to:
1 = √6/2
Since √6/2 = sin 15°, we have proven that sin 15° = (√6-√2)/4.