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Find the positive integers p, q, r, s sa...

Find the positive integers p, q, r, s satisfying `tan (pi/24)=(sqrt(p)-sqrt(q))(sqrt(r)-s)`

A

6

B

7

C

8

D

9

Text Solution

Verified by Experts

The correct Answer is:
C
Using `tan 2A = (2 tan A)/(1-tan^(2)A)`
`tan 7.5^(@)=(sin 15^(@))/(1+cos 15^(@))`
`=(sin 15^(@))/(1+cos 15^(@))xx(1-cos 15^(@))/(1-cos 15^(@))`
`=(sin 15^(@)(1-cos 15^(@)))/(sin^(2)15^(@))`
`=(1-cos 15^(@))/(sin 15^(@))`
`=(1-(sqrt(3)+1)/(2sqrt(2)))/((sqrt(3)-1)/(2sqrt(2)))`
`=(2sqrt(2)-sqrt(3)-1)/(sqrt(3)-1)`
`=(2sqrt(2)-sqrt(3)-1)/(sqrt(3)-1)xx(sqrt(3)+1)/(sqrt(3)+1)`
`=((2sqrt(2)-sqrt(3)-1)(sqrt(3)+1))/(3-1)`
`=(2sqrt(6)+2sqrt(2)-4-2sqrt(3))/(2)`
`=(2(sqrt(6)+sqrt(2)-2-sqrt(3)))/(2)`
`=sqrt(3)(sqrt(2)-1)-sqrt(2)(sqrt(2)-1)`
`= (sqrt(3)-sqrt(2))(sqrt(2)-1)`
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