Home
Class 12
MATHS
If a1, a2,a3, ,an is an A.P. with commo...

If `a_1, a_2,a_3, ,a_n` is an A.P. with common difference `d ,` then prove that `"tan"[tan^(-1)(d/(1+a_1a_2))+tan^(-1)(d/(1+a_2a_3))+......+tan^(-1)(d/(1+a_(n-1)a_n))]=((n-1)d)/(1+a_1a_n)`

Text Solution

Verified by Experts

We have
`tan^(-1) ((d)/(1 + a_(1) a_(2))) + tan^(-1) ((d)/(1 + a_(2) a_(3))) + ...+ tan^(-1) ((d)/(1 + a_(n -1) a_(n)))`
`= tan^(-1) ((a_(2) -a_(1))/(1 + a_(1) a_(2))) + tan^(-1) ((a_(3) -a_(2))/(1 + a_(2) a_(3))) + ...+ tan^(-1) ((a_(n) - a_(n -1))/(1 + a_(n -1) a_(n)))`
`= (tan^(-1) a_(2) - tan^(-1) a_(1)) + (tan^(-1) a_(3) - tan^(-1) a_(2)) + ...+ (tan^(-1) a_(n) - tan^(-1) a_(n))`
`= tan^(-1) a_(n) - tan^(-1) a_(1) = tan^(-1) ((a_(n) -a_(1))/(1 + a_(n) a_(1))) = tan^(-1) (((n -1))/(1 + a_(1) a_(n)))`
`rArr tan[tan^(-1) ((d)/(1 + a_(1) a_(2))) + tan^(-1) ((d)/(1 + a_(2) a_(3))) + ... + tan^(-1) ((d)/(1 + a_(n -1) a_(n)))] = ((n -1)d)/(1 + a_(1) a_(n))`
Promotional Banner

Topper's Solved these Questions

  • INTRODUCTION TO VECTORS

    CENGAGE PUBLICATION|Exercise TRUE OR FALSE|1 Videos

  • JEE 2019

    CENGAGE PUBLICATION|Exercise Chapter 10|8 Videos

Similar Questions

Explore conceptually related problems

If a_(1), a_(2), a_(3),…, a_(n) be in A.P. Show that, (1)/(a_(1)a_(2)) + (1)/(a_(2)a_(3)) +….+(1)/(a_(n-1)a_(n)) = (n-1)/(a_(1)a_(n))

Prove that: tan^(-1)(1/2tan2A)+tan^(-1)(cotA)+tan^(-1)(cot^3A) =0

If a_1,a_2,a3,...,a_n are in A.P then show that 1/(a_1a_2)+1/(a_2a_3)+1/(a_3a_4)+...+1/(a_(n-1)a_n)=(n-1)/(a_1a_n)

If a_1, a_2,...... ,a_n >0, then prove that (a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)+.....+(a_(n-1))/(a_n)+(a_n)/(a_1)> n

Prove that sum_(r=1)^(n) tan^(-1) ((2^(r -1))/(1 + 2^(2r -1))) = tan^(-1) (2^(n)) - (pi)/(4)

Evaluate: ("lim")_(n→oo)x[tan^(-1)((x+1)/(x+2))-tan^(-1)(x/(x+2))]

If alpha_1 , alpha_2, ......, alpha_n are in A.P, whose common difference is d , show that sin d [ sec alpha_1 sec alpha_2 + sec alpha_2 sec alpha_3 +.......+ sec alpha_(n - 1) sec alpha_n] = tan alpha_n - tan alpha_1

If a ,a_1, a_2, a_3, a_(2n),b are in A.P. and a ,g_1,g_2,g_3, ,g_(2n),b . are in G.P. and h s the H.M. of aa n db , then prove that (a_1+a_(2n))/(g_1g_(2n))+(a_2+a_(2n-1))/(g_1g_(2n-1))++(a_n+a_(n+1))/(g_ng_(n+1))=(2n)/h

If a_1,a_2, a_3, a_4 be the coefficient of four consecutive terms in the expansion of (1+x)^n , then prove that: (a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot

Prove that: tan^-1((c_1x-y)/(c_1y+x))+tan^-1((c_2-c_1)/(1+c_2c_1))+tan^-1((c_3-c_2)/(1+c_3c_2)) + .......+tan^-1((c_n-c_(n-1))/(1+c_nc_(n-1)))+tan^-1(1/c_n)=tan^-1(x/y)

CENGAGE PUBLICATION-INVERSE TRIGONOMETRIC FUNCTIONS-All Questions
  1. If x > y > z >0, then find the value of cot^(-1)((x y+1)/(x-y))+cot^(-...

    Text Solution

    |

  2. Solve tan^(-1) x + cot^(-1) (-|x|) = 2 tan^(-1) 6x

    Text Solution

    |

  3. If a1, a2,a3, ,an is an A.P. with common difference d , then prove th...

    Text Solution

    |

  4. Find the value of sum(r=0)^ootan^(-1)(1/(1+r+r^2))

    Text Solution

    |

  5. Find the sum sum(r =1)^(oo) tan^(-1) ((2(2r -1))/(4 + r^(2) (r^(2) -2r...

    Text Solution

    |

  6. Find the value of 4tan^(-1)(1/5)-tan^(-1)(1/(70))+tan^(-1)(1/(99))

    Text Solution

    |

  7. If (x -1) (x^(2) + 1) gt 0, then find the value of sin((1)/(2) tan^(-1...

    Text Solution

    |

  8. Prove that 3 tan^(-1) x= {(tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if ...

    Text Solution

    |

  9. Prove that cot^(-1).(3)/(4) + sin^(-1).(5)/(13) = sin^(-1).(63)/(65)

    Text Solution

    |

  10. Solve sin^(-1)x+sin^(-1)2x=pi/3dot

    Text Solution

    |

  11. Solve sin^(-1) x + sin^(-1) (1 - x) = cos^(-1) x

    Text Solution

    |

  12. cos^(-1){1/2x^(2)+sqrt(1-x^(2))sqrt(1-(x^(2))/(4))}=cos^(-1)((x)/(2))-...

    Text Solution

    |

  13. If x in (0,pi/2), then show that cos^(-1)(7/2(1+cos2x)+sqrt((sin^2x-48...

    Text Solution

    |

  14. Which of the following angles is greater? theta1=sin^(-1)+sin^(-1)1/3o...

    Text Solution

    |

  15. Find the value (lim)(nvecoo)sum(k=2)^n((1+sqrt((k-1)k(k+1)(k+2)))/(k(k...

    Text Solution

    |

  16. If f(x) = sin^(-1) x then prove that lim(x rarr (1^(+))/(2)) f(3x -4x...

    Text Solution

    |

  17. Solve sin^(-1)x-cos^(-1)x=sin^(-1)(3x-2)

    Text Solution

    |

  18. If A = 2 tan^(-1) ( 2 sqrt2 -1) " and " B = 3 sin ^(-1) ( 1/3) + sin^(...

    Text Solution

    |

  19. Prove that sin^(-1)((2x)/(1+x^2))=tan^(-1)((2x)/(1-x^2))

    Text Solution

    |

  20. If sin^(-1)((4x)/(x^2+4))+2tan^(-1)(-x/2) is independent of x , find t...

    Text Solution

    |