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If tan1^(@)tan2^(@)tan3^(@)tan4^(@). . ....

If `tan1^(@)tan2^(@)tan3^(@)tan4^(@). . .tan89^(@)=x^(2)-8`, then find the value of x.

Text Solution

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`x^(2)-8=(tan1^(@)tan89^(@))(tan2^(@)tan88^(@))(tan3^(@)tan87^(@)) . . .(tan44^(@)tan46^(@))tan45^(@)`
`=[tan1^(@)tan(90^(@)-1^(@))][tan2^(@)tan(90^(@)-2^(@))][tan3^(@)tan(90^(@)-3^(@))]`
`. . .[tan44^(@)tan(90^(@)-44^(@))]xx1`
`=(tan1^(@)cot1^(@))(tan2^(@)cot2^(@))(tan3^(@)cot3^(@)) . . . (tan44^(@)cot44^(@))`
`=1xx1xx1xx . . .xx1` (`:'tanx*cotx=*(1)/(tanx)=1`)
`rArrx^(2)-8=1`
`rArrx^(2)=9rArrx=+-3`
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