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सिद्ध करे की `tan1^(@).tan2^(@)...tan89^(@)=1`

Answer» (i) L.H.S `= tan 1^(@). tan 2^(@)....tan 89^(@)`
`=(tan 1^(@). tan 89^(@)).(tan 2^(@). tan 88^(@)...(tan 44^(@).tan 46^(2)).tan 45^(@)`
`=(tan 1^(@). cot 1^(@))(tan2^(@). cot2^(@))...(tan 44^(2).cot(44^(@).cot 44^(@)).tan 45^(@)`
`1xx1xx1xx..xx1=1=` R.H.S
(ii) L.H.S `=sin^(2) 5^(@) +sin^(2) 10^(@)+sin^(2)15^(@)+..+sin^(2)90^(@)`
`=(sin^(2) 5^(@)+sin^(2) 85^(@))+(sin^(2)10^(@) +sin^(2)80^(@))+...+(sin^(2) 40^(@)+sin^(2)50^(@)+sin^(2)45^(@)+sin^(2)90^(@)`
`=(sin^(2) 5^(@)+cos^(2) 5^(@))+(sin^(2) 10^(@))+...+(sin^(2)40^(@)+cos^(2) 40^(@)+sin^(2) 45^(@)+sin^(2)90^(@)`
`=(1+1+1+1+1+1+1)+(1)/sqrt(2)+1=9+1/2=91/2`


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