Question:
prove that : 2tan-1 3/4 - tan-1 17/34 = pi/4
Answer:
We have to prove that
2 tan-1 (3/4) - tan-1 (17/31) = π/4
We know that 2 tan-1 x = tan-1 {2x/(1 - x2 )}
So, 2 tan-1 x = tan-1 [(2*3/4)/{(1 - (3/4)2 }]
=> 2 tan-1 x = tan-1 [(3/2)/{1 - 9/16 }]
=> 2 tan-1 x = tan-1 [(3/2)/{(16 - 9)/16}]
=> 2 tan-1 x = tan-1 {(3/2)/(7/16)}
=> 2 tan-1 x = tan-1 {(3/2) * (16/7)}
=> 2 tan-1 x = tan-1 {3 * (8/7)}
=> 2 tan-1 x = tan-1 (24/7)
Now, 2 tan-1 (3/4) - tan-1 (17/34) = tan-1 (24/7) - tan-1 (17/31)
=> 2 tan-1 (3/4) - tan-1 (17/34) = tan-1 [{(24/7) - (17/31)}/{1 + (24/7) * (17/31)}]
=> 2 tan-1 (3/4) - tan-1 (17/34) = tan-1 [{(24*31 - 17*7)/(31*7)}/{1 + 408/(31*7)}]
=> 2 tan-1 (3/4) - tan-1 (17/34) = tan-1 {(744 - 119)/217}/{1 + 408/217}]
=> 2 tan-1 (3/4) - tan-1 (17/34) = tan-1 {(744 - 119)/217}/{(217 + 408)/217}]
=> 2 tan-1 (3/4) - tan-1 (17/34) = tan-1 {(744 - 119)/(217 + 408)}
=> 2 tan-1 (3/4) - tan-1 (17/34) = tan-1 (625/625)
=> 2 tan-1 (3/4) - tan-1 (17/34) = tan-1 1
=> 2 tan-1 (3/4) - tan-1 (17/34) = π/4
So, 2 tan-1 (3/4) - tan-1 (17/34) = π/4
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