Question

If sin (A+B) = sin A cos B + cos A sin B and cos (A-B) = cos A cos B + sin A sin B
(i) sin (75⁰)
(ii) cos (15⁰) 

Answer 1

Let A = 450 and B = 300
(i) As, sin(A + B) = sin A cos B + cos A sin B
⇒ sin (450 + 300)  = sin 450 cos 300 + cos 450 sin 300 

⇒ sin `(75^0) = 1/sqrt(2) xx sqrt(3)/2 + 1/sqrt(2) xx1/2`

⇒ sin `(75^0) = sqrt(3)/(2sqrt(2)) + 1/(2sqrt(2))`

∴ sin `(75^0) = (sqrt(3) +1)/(2 sqrt(2))`

ii) As, cos (A – B) = cos A cos B + sin A sin B
⇒ cos (45⁰ – 30⁰) = cos 45⁰ cos 30⁰ + sin 45⁰ sin 30⁰ 

⇒ cos `(15^0) = 1/sqrt(2) xx sqrt(3)/2 + 1/sqrt(2) xx1/2`

⇒ cos `(15^0) = sqrt(3)/(2sqrt(2)) + 1/(2sqrt(2))`

∴ cos `(15^0)=(sqrt(3)+1)/(2sqrt(2))`

Disclaimer: cos 15⁰ can also be written by taking A = 60⁰ and B = 45⁰.