if A1 A2 A3.... An are in ap with CD d then the sum of series:
sind[cosec A1*cosecA2+ cosecA2.cosecA3..........cosecA(n-1).cosecAn] is what
plz ans it correctly
sivaprasath:
What is the value of d ? in Q1??
its simply d
ok
what does CD mean there? in Q1??
common difference...u dont know what is AP
d = CD ??
yaa so what cant common difference b in any variable
actually u dont know
ans will be in variable
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A₁ , A₂ , A₃ , .......An are in AP and common difference is d
so, d = A₂ - A₁ = A₃ - A₂ = A₄ - A₃ =...... =An - An₋₁ -------(1)
Now, sind[cosecA₁.cosecA₂ + cosecA₂.cosecA₃ + cosecA₃.cosecA₄ ...........+ cosecAn₋₁.cosecAn ]
⇒sind[1/sinA₁.sinA₂ + 1/sinA₂.sinA₃ + 1/sinA₃.sinA₄ + .........+ 1/sinAn₋₁.sinAn]
⇒sind/sinA₁.sinA₂ + sind/sinA₂.sinA₃ + sind/sinA₃.sinA₄ + .......+sind/sinAn₋₁.sinAn
⇒sin(A₂ - A₁)/sinA₁.sinA₂ + sin(A₃ - A₂)/sinA₂.sinA₃ + sin(A₄ - A₃)/sinA₃.sinA₄ + .......+ sin(An₋₁ - An)/sinAn₋₁.sinAn [ from equation (1)
⇒(sinA₁.cosA₂ - cosA₁.sinA₂)/sinA₁.sinA₂ + (sinA₂.cosA₃ - cosA₂.sinA₃)/sinA₂.sinA₃ +.........+ (sinAn₋₁.cosAn - cosAn₋₁.sinAn)/sinAn₋₁.sinAn
⇒cotA₂ - cotA₁ + cotA₃ - cotA₂ + cotA₄ - cotA₃ + .........cotAn - cotAn₋₁
⇒cotAn- cotA₁
so, d = A₂ - A₁ = A₃ - A₂ = A₄ - A₃ =...... =An - An₋₁ -------(1)
Now, sind[cosecA₁.cosecA₂ + cosecA₂.cosecA₃ + cosecA₃.cosecA₄ ...........+ cosecAn₋₁.cosecAn ]
⇒sind[1/sinA₁.sinA₂ + 1/sinA₂.sinA₃ + 1/sinA₃.sinA₄ + .........+ 1/sinAn₋₁.sinAn]
⇒sind/sinA₁.sinA₂ + sind/sinA₂.sinA₃ + sind/sinA₃.sinA₄ + .......+sind/sinAn₋₁.sinAn
⇒sin(A₂ - A₁)/sinA₁.sinA₂ + sin(A₃ - A₂)/sinA₂.sinA₃ + sin(A₄ - A₃)/sinA₃.sinA₄ + .......+ sin(An₋₁ - An)/sinAn₋₁.sinAn [ from equation (1)
⇒(sinA₁.cosA₂ - cosA₁.sinA₂)/sinA₁.sinA₂ + (sinA₂.cosA₃ - cosA₂.sinA₃)/sinA₂.sinA₃ +.........+ (sinAn₋₁.cosAn - cosAn₋₁.sinAn)/sinAn₋₁.sinAn
⇒cotA₂ - cotA₁ + cotA₃ - cotA₂ + cotA₄ - cotA₃ + .........cotAn - cotAn₋₁
⇒cotAn- cotA₁
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