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Question
If sin(A - B) = sinA cosB - cosA sinB and cos(A - B) = cosA cosB + sinA sinB, find the values of sin15° and cos15°.
Solution
Let A = 45° and B = 30°
Then,
sin(A - B) = sinA cosB - cosA sinB
⇒ sin45° - 30°) = sin45° cos30° - cos45° sin30°
⇒ sin15° = `(1)/sqrt(2) xx sqrt(3)/(2) - (1)/sqrt(2) xx (1)/sqrt(2)`
⇒ sin15° = `sqrt(3)/(2sqrt(2)) - (1)/(2sqrt(2)`
⇒ sin15° = `((sqrt(3) - 1))/(2sqrt(2)`
cos(A -B) = cosA cosB + sinA sinB
⇒ cos(45° - 30°) = cos45° cos30° + sin45° sin30°
⇒ cos15° = `(1)/sqrt(2) xx sqrt(3)/(2) + (1)/sqrt(2) xx (1)/sqrt(2)`
⇒ cos15° = `sqrt(3)/(2sqrt(2)) + (1)/(2sqrt(2)`
⇒ cos15° = `((sqrt(3) + 1))/(2sqrt(2)`.
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