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Prove that : (i) tan 5^@tan 25^@ tan 3...

Prove that :
`(i) tan 5^@tan 25^@ tan 30^@tan 65^@tan 85^@=(1)/(sqrt(3))`
`(ii) cot 12^@cot38^@cot 52^@cot60^@cot78^@=(1)/(sqrt(3))`
`(iii) cos 15^@cos 35^@cosec 55^@cos 60^@cosec 75^@=(1)/(2)`
`(iv) cos 1^@cos 2^@cos 3^@.....cos 180^@=0`
`(v) ((sin 49^@)/(cos 41^@))^2+ ((cos 41^@)/(sin 49^@))^2=2`

Text Solution

Verified by Experts

`(i) LHS =(tan 5^@tan 85^@)(tan 25^@tan 65^@)tan 30^@`.
`(ii) LHS=(cot 12^@cot 78^@)(cot 38^@cot 52^@)cot 60^@`.
`(iii) LHS =(cos 15^@cosec 75^@(cot 38^@cot 55^@) cos 60^@`.
`(iv) LHS =cos 1^@cos 2^@cos 3^@......cos 89^@cos 90^@cos 91^@.....cos 180^@=0, "since "cos 90^@=0`.
`(v) LHS ={(sin 49^@)/(cos(90^@-49^@))}^2+{(cos 41^@)/(sin(90^@-41^@))}^2=((sin49^@)/(sin41^@))^2+((cos41^@)/(cos41^@))^2=1+1=2`.
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