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Find the exact value of the expression ...

Find the exact value of the expression
`T=(sin 40^(@))/(sin80^(@))+(sin80^(@))/(sin20^(@))-(sin20^(@))/(sin40^(@))`.

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
C
`(1)/(2 cos 40^(@))+4 cos 40^(@). Cos 20^(@)-(1)/(2 cos 20^(@))`
`=(1)/(2)[(1)/(cos 40^(@))-(1)/(cos 20^(@))]+2(cos 60^(@)+cos 20^(@))`
`=(1)/(2)[(cos 20^(@)-cos 40^(@))/(cos 40^(@)cos 20^(@))]+1+2 cos 20^(@)`
`=(2 sin 30^(@)sin10^(@))/(2 cos 40^(@) cos20^(@))+1+2 cos 20^(@)`
`=(2sin 10^(@)sin 20^(@))/(sin 80^(@))+1+2 cos 20^(@)`
`=(2sin 10^(@)sin 20^(@)+2 cos 20^(@) cos 10^(@))/(sin 80^(@))+1`
`=(cos 10^(@)-cos 30^(@)+cos 30^(@)+cos 10^(@))/(sin 80^(@))+1`
`=(2 cos 10^(@))/(sin 80^(@))+1=3`
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