Home
Class 10
MATHS
Prove that (i) (1- sin theta)/(1+ sin ...

Prove that
`(i) (1- sin theta)/(1+ sin theta)=(sec theta - tan theta )^(2) `
`(ii) ((1+cos theta))/((1- cos theta))= ("cosec" theta + cot theta)^(2)`

Text Solution

Verified by Experts

We have
`(i) RHS = (sec theta - tan theta)^(2) `
` =((1)/(cos theta)-(sin theta)/(cos theta))^(2)=((1-sin theta)/(cos theta))^(2)=((1- sin theta)^(2))/(cos^(2)theta)`
` = ((1-sin theta)^(2))/((1-sin^(2)theta))=((1- sin theta)(1-sin theta))/((1+ sin theta)(1-sin theta))`
`=((1-sin theta))/((1+sin theta))= LHS. `
` therefore RHS = LHS. `
`(ii) RHS= ("cosec" theta + cot theta)^(2) =((1)/(sin theta)+(cos theta)/(sin theta))^(2) `
` =((1+ cos theta)/(sin theta))^(2) = ((1+ cos theta)^(2))/(sin^(2)theta) =((1+ cos theta)(1+ cos theta))/(1- cos^(2)theta)`
` =((1+ cos theta)(1+ cos theta))/((1+ cos theta)(1- cos theta))=((1+ cos theta))/((1- cos theta))= LHS. `
`therefore LHS = RHS. `
Promotional Banner

Topper's Solved these Questions

  • TRIGONOMETRIC IDENTITIES

    RS AGGARWAL|Exercise Exercise 13A|38 Videos

  • TRIGONOMETRIC IDENTITIES

    RS AGGARWAL|Exercise Exercise 13B|15 Videos

  • TRIANGLES

    RS AGGARWAL|Exercise Test Yourself|20 Videos

  • TRIGONOMETRIC RATIOS

    RS AGGARWAL|Exercise Exercise 10|35 Videos

Similar Questions

Explore conceptually related problems

Prove each of the following identities : (i) sqrt((1+ sin theta)/(1- sin theta)) = ( sec theta + tan theta) (ii) sqrt((1- cos theta)/(1+ cos theta))= ("cosec" theta - cot theta) (iii) sqrt((1+ cos theta)/(1- cos theta))+ sqrt((1- cos theta)/(1+ cos theta)) = 2 "cosec" theta

Prove the following trigonometric identities: sqrt((1-sin theta)/(1+sin theta))=sec theta-tan thetasqrt((1+cos theta)/(1-cos theta))=csc theta+cot theta

Prove that (1+sin theta-cos theta)/(1+sin theta+cos theta)=tan((theta)/(2))

Prove that: :(1+sin theta-cos theta)/(1+sin theta+cos theta)=tan(theta)/(2)

Prove that (i) (sin theta)/((1- cos theta))=("cosec" theta + cot theta) (ii) (1)/((sec theta - tan theta))=(sec theta + tan theta)

Prove that (cos theta)/(1+sin theta)=(1-sin theta)/(cos theta)=sec theta-tan theta

Prove that (sec theta+tan theta)(1-sin theta)=cos theta

Prove each of the following identities : (i) sec theta(1- sin theta)(sec theta + tan theta)=1 (ii) sin theta (1+ tan theta) + cos theta( 1+ cot theta)=( sec theta + "cosec" theta)

Prove that (cos theta-sin theta+1)/(cos theta+sin theta-1)=csc theta+cot theta

Prove that: (sin theta)/(1 - cos theta) + (tan theta)/(1 + cos theta) = cosec theta + cot theta + sec theta cosec theta - cosec theta

RS AGGARWAL-TRIGONOMETRIC IDENTITIES-Test Yourself
  1. Prove that (i) (1- sin theta)/(1+ sin theta)=(sec theta - tan theta ...

    Text Solution

    |

  2. (cos^(2)56^(@)+cos^(2)34^(@))/(sin^(@)56^(@)+sin^(2)34^(@))+3tan^(2)56...

    Text Solution

    |

  3. (sin^2 30^@ cos^2 45^@+4 tan^2 30^@+1/2 sin^2 90^@+1/8 cot^2 60^@)=?

    Text Solution

    |

  4. If cosA+cos^(2)A=1, then sin^(2)A+sin^(4)A=?.

    Text Solution

    |

  5. If sin theta=(sqrt(3))/(2)" then " ("cosec"theta+cot theta)=?

    Text Solution

    |

  6. If cotA=(4)/(5), then ((sinA+cosA))/((sinA-cosA))=?

    Text Solution

    |

  7. If 2x=secA and (2)/(x)=tanA, " prove that " (x^(2)-(1)/(x^(2)))=(1)/(4...

    Text Solution

    |

  8. If sqrt(3) tantheta=3sintheta, " prove that " (sin^(2)theta-cos^(2)the...

    Text Solution

    |

  9. Prove that ((sin^(2)73^(@)+sin^(2)17^(@)))/((cos^(2)28^(@)+cos^(2)62^(...

    Text Solution

    |

  10. If 2sin2 theta=sqrt(3) ,then the value of theta is :

    Text Solution

    |

  11. Prove that sqrt((1+cosA)/(1-cosA))=("cosec"A+cotA).

    Text Solution

    |

  12. If cosec theta + cot theta= p, cos theta = ?

    Text Solution

    |

  13. Prove that ("cosec"A-cotA)^(2)=((1-cosA))/((1+cosA)).

    Text Solution

    |

  14. If 5cottheta=3. find the value of (5sintheta-3costheta)/(4sintheta+3co...

    Text Solution

    |

  15. Prove that (sin32^(@)cos58^(@)+cos32^(@)sin58^(@))=1.

    Text Solution

    |

  16. If x=a sin theta+b cos theta and y=a cos theta -b sin theta, " prove t...

    Text Solution

    |

  17. ((1+sintheta))/((1-sintheta))=?

    Text Solution

    |

  18. Prove that (1)/((sec theta - tan theta))-(1)/(cos theta)=(1)/(cos th...

    Text Solution

    |

  19. ((sinA-2sin^(3)A))/((2cos^(3)A-cosA)) = ?

    Text Solution

    |

  20. Prove that (tanA)/((1-cotA))+(cotA)/((1-tanA))=(1+tanA+cotA).

    Text Solution

    |

  21. If sec5A="cosec"(A-36^(@)) and 5A " is an acute angle, show that " A=2...

    Text Solution

    |