Let
x=(1+tan1)(1+tan2)....(1+tan25)
y=(1-tan136)(1-tan137)....(1-tan160)
Then xy equals to
Answers
Explanation:
=(1+tan 0o)(1 + tan1°) (1 + tan2°) (1 + tan3°) ………. (1 + tan 45°) {Since , tan 0o = 0}
Now , we know that
tan(A + B) = [ tan(A) + tan(B) ] / [ 1 - tan(A) tan(B) ]
And tan(45) = 1
So
tan(45) = [ tan1 + tan44 ] / [ 1 - tan1 tan44 ]
tan(45) = [ tan2 + tan43 ] / [ 1 - tan2 tan43 ]
.
.
.
tan(45) = [ tan22 + tan23 ] / [ 1 - tan22 tan23 ]
So we have ,
tan1 + tan44 = tan(45) [ 1 - tan1 tan44 ]
tan2 + tan43 = tan(45) [ 1 - tan2 tan43 ]
.
.
.
tan22 + tan23 = tan(45) [ 1 - tan22 tan23 ]
So finally we have ,
(1 + tan1) (1 + tan2) (1 + tan3) ... (1 + tan44) (1 + tan45)
= (1 + tan1) (1 + tan2) (1 + tan3) ... (1 + tan44) (1 + 1)
= 2 (1 + tan1) (1 + tan2) (1 + tan3) ... (1 + tan44)
= 2 (1 + tan1) (1 + tan44) * (1 + tan2) (1 + tan43) * (1 + tan3) (1 + tan42) * ... * (1 + tan22) (1 + tan23)
= 2 (tan1 + tan44 + tan1 tan44 + 1) (tan2 + tan43 + tan2 tan43 + 1) ... (tan22 + tan23 + tan22 tan23 + 1)
= 2 ((tan45)(1 - tan1 tan44) + tan1 tan44 + 1) ((tan45)(1 - tan2 tan43) + tan2 tan43 + 1) ... ((tan45)(1 - tan22 tan23) + tan22 tan23 + 1)
= 2 (1 - tan1 tan44 + tan1 tan44 + 1) (1 - tan2 tan43 + tan2 tan43 + 1) ... (1 - tan22 tan23 + tan22 tan23 + 1)
= 2 (2) (2) ... (2)
= 2 (2^22)
= 2^23
Given : x=(1+tan1)(1+tan2)....(1+tan25) , y=(1-tan136)(1-tan137)....(1-tan160)
To find : xy
Solution:
x = (1+tan1)(1+tan2)........(1+tan25)
y = (1-tan136)(1-tan137)....(1-tan160)
tan 136 = -tan44 , tan137 = -tan43 ,.......tan160 = - tan20
=> y = ( 1 + tan44)(1 + tan43)...................(1 + tan20)
xy = ((1+tan1)(1+tan2)........(1+tan25))(( 1 + tan44)(1 + tan43)...................(1 + tan20))
=> xy = (1 + tan1)(1 + Tan44)(1 + tan2)(1 + tan43) ....................(1 + tan25)((1 + tan20)
now using concept
Tan ( A + B) = (TanA + TanB )/(1 - TanATanB) such that A + B = 45°
=> 1 = (TanA + TanB )/(1 - TanATanB)
=> 1 - TanATanB = TanA + TanB
=> 1 = TanA + TanB + TanATanB
=> 1 + 1 = 1 + TanA + TanB + TanATanB
=> 2 = ( 1 + TanA) + TanB(1 + TanA)
=> 2 = (1 + TanA)(1 + TanB)
=> xy = (2)(2) ....................................(2) ( 25 times)
=> xy = 2²⁵
xy = 2²⁵