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if x1, x2, x3......xn in AP then prove thatsecx1.secx2 + secx2.secx3 + secx3.secx4................................secxn-1.secxn = (tanxn-tanx1)/sin(x2-x1)

if x1, x2, x3......xn in AP then prove thatsecx1.secx2 + secx2.secx3 + secx3.secx4................................secxn-1.secxn = (tanxn-tanx1)/sin(x2-x1)

Grade:12th pass

1 Answers

Ritesh Khatri
76 Points
3 years ago
secxr.secxr+1
=1/(cosxr.cosxr+1)
= sind/(sind.cosxr.cosxr+1)
(where d is difference of given AP)
=sin(xr+1-xr)/(sind.cosxr.cosxr+1)
= (sinxr+1cosxr – cosxr+1sinxr)/(sind.cosxr.cosxr+1)
=[(sinxr+1cosxr)/(sind.cosxr.cosxr+1)] – [(cosxr+1sinxr)/(sind.cosxr.cosxr+1)]
=[(sinxr+1)/(sind.cosxr+1)] – [(sinxr)/(sind.cosxr.)]
=[tanxr+1 – tanxr]/sind
Now substitute r=1,2,3,....,n-1 and add, we will get
(tanxn – tanx1)/sind
=(tanxn – tanx1)/sin(x2 – x1)

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