Tan 75°+cot 75°=4

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Solution

tan 75°

= tan ( 45° + 30° )

= ( tan 45° + tan 30° ) / ( 1 - tan 45° tan 30° )

= [ 1 + (1/√3) ] / [ 1 - (1/√3) ]

= ( √3 + 1 ) / ( √3 - 1 )

tan 75° + cot 75°

= tan 75° + ( 1 / tan 75° )

= [ (√3 + 1 ) / ( √3 - 1 ) ] + [ (√3 - 1 ) / ( √3 + 1 ) ]

= [ (√3 + 1 )² + ( √3 - 1 )² ] / [ ( √3 - 1 )( √3 + 1 ) ]

= 2 [ (√3)² + 1² ] / [ (√3)² - 1² ]

= 2 [ 3 + 1 ] / [ 3 - 1 ]

= 4

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