Home
Class 11
MATHS
Prove that (cos3x)/(sin2xsin4x)+(cos5x)...

Prove that `(cos3x)/(sin2xsin4x)+(cos5x)/(sin4xsin6x)+(cos7x)/(sin6xsin8x)+(cos9x)/(sin8xsin10 x)` = `1/2(cos e cx)[cos e c2x-cos e c10x]`

Text Solution

Verified by Experts

`L.H.S. = (cos3x)/(sin2xsin4x)+(cos5x)/(sin4xsin6x)+(cos7x)/(sin6xsin8x)+(cos9x)/(sin8xsin10x)`
`=1/(2sinx)[ (2sinxcos3x)/(sin2xsin4x)+(2sinxcos5x)/(sin4xsin6x)+(2sinxcos7x)/(sin6xsin8x)+(2sinxcos9x)/(sin8xsin10x)]`
`=1/(2sinx)[ (2sinxcos3x)/(sin2xsin4x)+(2sinxcos5x)/(sin4xsin6x)+(2sinxcos7x)/(sin6xsin8x)+(2sinxcos9x)/(sin8xsin10x)]`
Using `2cosAsinB = sin(A+B) - sin(A-B)`
`=1/(2sinx)[ (sin4x-sin2x)/(sin2xsin4x)+(sin6x-sin4x)/(sin4xsin6x)+(sin8x-sin6x)/(sin6xsin8x)+(sin10x-sin8x)/(sin8xsin10x)]`
`=1/(2sinx)[ 1/(sin2x) - 1/(sin4x)+1/(sin4x) - 1/(sin6x)+1/(sin6x) - 1/(sin8x)+1/(sin8x) - 1/(sin10x)]`
`=1/(2sinx)[1/(sin2x) - 1/(sin10x)]`
`=1/2cosecx[cosec2x-cosec10x] = R.H.S.`
Promotional Banner

Topper's Solved these Questions

  • STRAIGHT LINES

    CENGAGE|Exercise Solved Examples And Exercises|381 Videos

Similar Questions

Explore conceptually related problems

(cos2x)/(sin ^(2) x cos^(2) x)

Prove that (cos 7x + cos 5x)/( sin 7x - sin 5x) = cot x

Prove that: (sinx)/(cos3x)+(sin3x)/(cos9x)+(sin9x)/(cos27 x)=(1/2)(tan27 x-tanx)

Prove that (sin4x+sin2x)/(cos4x+cos2x)=tan3x

Prove that (sinx+sin3x+sin5x+sin7x)/(cosx+cos3x+cos5x+cos7x)=tan4x

Prove that sqrt(sin^4x+4cos^2x)-sqrt(cos^4x+4sin^2x)=cos2xdot

(3 + 4 cos x)/( sin^(2) x )

lim_(xto0)(1-cos^(2)x)/(xsin2x)

Prove that (cos 3x +cos 7x - cos 2x )/( sin 7x - sin 3x - sin 2x ) = cot 2x

Prove that: 3 (sin x-cos x)^4+ 6 (sin x +cosx)^ 2+4 (sin^6 x+ cos^6 x) -13=0

CENGAGE-TRIGONOMETRIC FUNCTIONS-Solved Examples And Exercises
  1. Find the value of cos(2pi)/7+cos(4pi)/7+cos(6pi)/7

    Text Solution

    |

  2. Prove that sintheta+sin3theta+sin5theta+.....+sin(2n-1)theta=(sin^2nth...

    Text Solution

    |

  3. Prove that (cos3x)/(sin2xsin4x)+(cos5x)/(sin4xsin6x)+(cos7x)/(sin6xsi...

    Text Solution

    |

  4. Prove that 2sin2^0+4sin4^0+6"sin"6^0+.......+180sin180^0=90cot1^0dot

    Text Solution

    |

  5. If A+B+C=pi prove that cos^(2) A+cos^(2) B+cos^(2) C=1" cos A cos B co...

    Text Solution

    |

  6. Prove that in triangle A B C ,cos^2A+cos^2B-cos^2C=1-2sinAsinBcosCdot

    Text Solution

    |

  7. In triangle ABC, prove that sin(B+C-A)+sin(C+A-B)+sin(A+B-C)=4sin Asi...

    Text Solution

    |

  8. Prove that sum(k=1)^(n-1)(n-k)cos(2kpi)/n=-n/2, wherengeq3 is an int...

    Text Solution

    |

  9. If (tan(ln6)dottan(ln2)dot"tan"(ln3))/(tan(ln6)-tan(ln2)-"tan"(ln3))=k...

    Text Solution

    |

  10. In any triangle A B C ,sin^2A-sin^2B+sin^2C is always equal to (A) 2s...

    Text Solution

    |

  11. If cot^2Acot^2B=3, then the value of (2-cos2A)(2-cos2B) is

    Text Solution

    |

  12. If tanalpha=m/(m+1) and tanbeta=1/(2m+1) . Find the possible values o...

    Text Solution

    |

  13. If u = sqrt( a^2cos^2 theta +b^2 sin ^2 theta ) + sqrt( a^2 sin^2 ...

    Text Solution

    |

  14. The value of sin^2 12^0+sin^2 21^0+sin^2 39^0+sin^2 48^0-sin^2 9^0-sin...

    Text Solution

    |

  15. The value of f(x)=x^4+4x^3+2x^2-4x+7, when x=cot((11pi)/8) is

    Text Solution

    |

  16. Find the value of sqrt3" cosec "20^(@)-sec 20^(@)

    Text Solution

    |

  17. If (1+sint)(1+cost)=5/4 then find the value of (1-sint)(1-cost)dot

    Text Solution

    |

  18. If alpha,beta,gamma,delta are the smallest positive angles in ascendi...

    Text Solution

    |

  19. (sin^2A-sin^2B)/(sinAcosA-sinBcosB)is equal to (a)"tan"(A-B) ...

    Text Solution

    |

  20. If cos25^0+sin25^0=p , then cos50^0 is (a)sqrt(2-p^2) (b) -sqrt(2...

    Text Solution

    |