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Prove that (1+ tan 1^@ ) (1+ tan 2^@ ).....

Prove that `(1+ tan 1^@ ) (1+ tan 2^@ )...(1+ tan 45º )= 2^23`

Text Solution

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`(1+tanx^(@))(1+tan($5^(@)-x^(@)))`
`=(1+tanx^(@))(1+tan 44^(@))=(1+tan2^(@))(1+tan43^(@))`
`=(1+tan3^(@))(1+tan42^(@))`
`=(1+tan 22^(@))(1+tan23^(@))`
`=2`
`(1+tan1^(@))(1+tan2^(@)).....(1+tan 45^(@))=2^(23)`
(as `1+tan 45^(@)=2`)
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