Home
Class 12
MATHS
सिद्ध कीजिए tan^(-1)sqrt(x)=(1)/(2)co...

सिद्ध कीजिए
`tan^(-1)sqrt(x)=(1)/(2)cos^(-1) ""((1-x)/(1+x)), x in [0,1]`

Text Solution

Verified by Experts

Putting, `x=tan^(2)theta`, we get
RHS `=1/2cos^(-1)(1-x)/(1+x)`
`=1/2cos^(-1)((1-tan^(2)theta)/(1+tan^(2)theta))`
`=1/2cos^(-1)(cos2theta)`
`=(1/2 xx 2theta) = theta=tan^(-1)sqrt(x)`= LHS
`[therefore x=tan^(2)theta rArr tantheta=sqrt(x) rArr theta=tan^(-1)sqrt(x)]`
Hence, `tan^(-1)=1/2cos^(-1)((1-x)(1+x))`.
Promotional Banner

Topper's Solved these Questions

  • INVERSE TRIGNOMETRIC FUNCTIONS

    RS AGGARWAL|Exercise Exercise 4A|4 Videos

  • INVERSE TRIGNOMETRIC FUNCTIONS

    RS AGGARWAL|Exercise Exercise 4B|22 Videos

  • INTEGRATION USING PARTIAL FRACTIONS

    RS AGGARWAL|Exercise Objective Questions Ii|37 Videos

  • LINEAR DIFFERENTIAL EQUATIONS

    RS AGGARWAL|Exercise Objective Questions|27 Videos

Similar Questions

Explore conceptually related problems

Toprove tan ^(-1)sqrt(x)=(1)/(2)cos^(-1)((1-x)/(1+x))

prove that tan^(-1)(sqrt(x))=(1)/(2)cos^(-1)((1-x)/(1+x)),x in[0,1]

Prove that: quad tan^(-1)sqrt(x)=(1)/(2)cos^(-1)((1-x)/(1+x)),x in[0,1]

सिद्ध कीजिए कि 2tan^(-1)sqrt((b)/(a))=cos^(-1)((a-b)/(a+b))

tan^(-1)(sqrt((a-x)/(a+x)))=(1)/(2)cos^(-1)((x)/(a))

Prove the following: tan^(-1)sqrt(x)=(1)/(2)cos^(-1)((1-x)/(1+x)),x in(0,1)

tan^(-1)((1)/(sqrt(3))(tan x)/(2))=(1)/(2)cos^(-1)((1+2cos x)/(2+cos x))

tan ^(-1)(sqrt((1-cos 3 x)/(1+cos x)))

tan^(-1)((sqrt(1+x)+sqrt(1-x))/(sqrt(1+x)-sqrt(1-x)))

(sin^(-1)sqrt(x)-cos^(-1)sqrt(x))/(sin^(-1)sqrt(x)+cos^(-1)sqrt(x)),x in[0,1]

RS AGGARWAL-INVERSE TRIGNOMETRIC FUNCTIONS -Objective Questons
  1. सिद्ध कीजिए tan^(-1)sqrt(x)=(1)/(2)cos^(-1) ""((1-x)/(1+x)), x in [...

    Text Solution

    |

  2. Find the principal value of: cos^(-1)((sqrt(3))/2)

    Text Solution

    |

  3. Find the principal value of the following: (i) cosec^(-1)(2) (ii) ta...

    Text Solution

    |

  4. Find the principal value of: cos^(-1)(-1/(sqrt(2)))

    Text Solution

    |

  5. Find the principal value of: sin^(-1)(-1/2)

    Text Solution

    |

  6. The principal value of cos^(-1)(-1/2) is

    Text Solution

    |

  7. The principal value of tan^-1(-sqrt3) is

    Text Solution

    |

  8. The principal value of cot^(-1)(-1) is

    Text Solution

    |

  9. Find the principal value of sec^(-1)((-2)/(sqrt(3)))

    Text Solution

    |

  10. The principal value of "cosec"^(-1)(-sqrt(2)) is

    Text Solution

    |

  11. Find the principal values of each of the following: cot^(-1)(-sqrt(...

    Text Solution

    |

  12. The principle value of sin ^(-1)(sin ""(2pi)/(3)) is

    Text Solution

    |

  13. The value of cos^(-1)(cos(14pi/6)) is

    Text Solution

    |

  14. The value of tan^(-1)(tan(8*pi/6)) is

    Text Solution

    |

  15. Find the value of : cot^(-1), (5pi/4)

    Text Solution

    |

  16. The value of sec^(-1)(sec(8pi)/5) is

    Text Solution

    |

  17. The value of "cosec"^(-1)("cosec"(4pi)/3) is

    Text Solution

    |

  18. प्रश्न संख्या 16 से 18में दिए प्रत्येक व्यंजन का मान ज्ञात कीजिए : t...

    Text Solution

    |

  19. pi/3-sin^(-1)(-1/2)=?

    Text Solution

    |

  20. The value of sin(sin^(-1)(1/2)+cos^(-1)(1/2))=?

    Text Solution

    |

  21. If x ne 0 then cos(tan^(-1)+cot^(-1)x)=?

    Text Solution

    |