Evaluate tan^(-1) 2+tan^(-1) 2/9+ tan ^(-1) 2/25 tan^(-1)2/49+....upto n terms

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Please find below the solution to the asked query:

Let S=tan-12+tan-129+tan-1225+tan-1249+········upto n terms ;equationiWe know that  tan-1x+tan-1y=tan-1x+y1-xy ; x,y>0 and xy<1Using this identity in equation i , we getS=tan-12+291-2×29+tan-1225+tan-1249+········upto n terms S=tan-118+299-49+tan-1225+tan-1249+········upto n terms S=tan-1205+tan-1225+tan-1249+········upto n terms S=tan-14+tan-1225+tan-1249+········upto n termsS=tan-14+2251-4×225+tan-1249+········upto n termsS=tan-1100+22525-825+tan-1249+········upto n termsS=tan-110217+tan-1249+········upto n termsS=tan-16+tan-1249+········upto n termsNow if you observe When we take first term of S we get tan-12=tan-12×1When we add first two terms of S we get tan-14=tan-12×2When we add first three terms of S we get tan-16=tan-12×3SimilarlyWhen we add n terms of S we will get  tan-12×n=tan-12nS=tan-12+tan-129+tan-1225+tan-1249+········upto n terms= tan-12n

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