Question

If A+B=45 prove Then (cot A-1) (cot B-1)=2

Answer 1

A + B = 45°

take tan both sides

tan (A + B) = tan45°

{tanA + tanB } /(1 - tanA.tanB) = 1

tanA + tanB = 1 - tanA.tanB

1/cotA + 1/cotB = 1 - 1/cotA.cotB

(cotA + cotB ) = cotA.cotB - 1

cotA + cotB - cotA.cotB + 1 =0

cotA + cotB -cotA.cotB -1 = -2

cotA ( 1 - cotB ) + (cotB - 1) = - 2

(cotA - 1 )(1 - cotB ) = -2

(cotA -1 )( cotB -1 ) = 2

hence proved

Answer 2

Answer:

A + B = 45°

taking tan on both sides

,

tan (A + B) = tan45°

{tanA + tanB } /(1 - tanA.tanB) = 1

tanA + tanB = 1 - tanA.tanB

1/cotA + 1/cotB = 1 - 1/cotA.cotB

(cotA + cotB ) = cotA.cotB - 1

cotA + cotB - cotA.cotB + 1 =0

cotA + cotB -cotA.cotB -1 = -2

cotA ( 1 - cotB ) + (cotB - 1) = - 2

(cotA - 1 )(1 - cotB ) = -2

(cotA -1 )( cotB -1 ) = 2

Hence Proved

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