Question

The value of sin^2 75° - sin^2 15° is​

Answer 1

Solution:

sin²75° - sin²15°

It can be written as

sin75° = sin(30° + 45°)

sin15° = sin(45° - 30°)

Using , sin(A + B) = sinAcosB + cosAsinB

sin(A - B) = sinAcosB - cosAsinB

♦ (sin30°cos45° + cos30°sin45°)² - (sin30°cos45° - cos30°sin45°)

♦ (1/2 × 1/√2 + √3/2 × 1/√2)² - (1/2 × 1/√2 - √3/2 × 1/√2)²

♦ [(1 + √3)/2√2]² - [(1 - √3)/2√2]²

♦ (1 + √3)²/(2√2)² - { (1 - √3)²/(2√2)² }

Using

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

♦ [1² + (√3)² + 2(√3)(1)]/4(2)] - {[1² + (√3)² - 2(√3)(1)]/4(2)}

♦ 4 + 2√3/8 - ( 4 - 2√3/8)

Taking common

♦ 2(2 + √3)/8 - [ 2(2 - √3)/8 ]

♦ 2 + √3/4 - (2 - √3)/4

♦ 2 + √3/4 - 2 + √3/4

♦ 2 + √3 - 2 + √3/4

♦ 2√3/4

♦ √3/2

Hence , sin²75° - sin²15° = √3/2