Home
Class 10
MATHS
Prove the following identities: tan^2A-t...

Prove the following identities: `tan^2A-tan^2B=(cos^2B-cos^2A)/(cos^2Bcos^2A)=(sin^2A-sin^2B)/(cos^2Acos^2B)` `(sinA-sinB)/(cosA+cosB)+(cosA-cosB)/(sinA+sinB)=0`

Text Solution

Verified by Experts

We have
` (tan^(2)A - tan^(2)B) = (sin^(2)A)/(cos^(2)A) - (sin^(2)B)/(cos^(2)B) `
` = (sin^(2)A cos^(2)B - cos^(2)A sin^(2)B)/(cos^(2)A cos^(2)B) `
` = (sin^(2)A(1- sin^(2)B)-(1- sin^(2)A)sin^(2)B )/(cos^(2)A cos^(2)B) `
` = ((sin^(2)A - sin^(2)B))/(cos^(2)A cos^(2)B).`
Also, ` (tan^(2)A - tan^(2)B)= (sin^(2)A)/(cos^(2)A)- (sin^(2)B)/(cos^(2)B) `
` = (sin^(2)A cos^(2)B - cos^(2)A sin^(2)B)/(cos^(2)A cos^(2)B) `
` = ((1- cos^(2)A)cos^(2)B - cos^(2)A(1- cos^(2)B))/(cos^(2)B cos^(2)A) `
` = ((cos^(2)B - cos^(2)A))/(cos^(2)B cos^(2)A). `
Hence, `(tan^(2)A - tan^(2)B) = ((sin^(2)A - sin^(2)B))/(cos^(2)A cos^(2)B) = ((cos^(2)B - cos^(2)A))/(cos^(2)B cos^(2)A). `
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 the following identities: tan^(2)A-tan^(2)B=(cos^(2)B-cos^(2)A)/(cos^(2)B cos^(2)A)=(sin^(2)A-sin^(2)B)/(cos^(2)A cos^(2)B)(sin A-sin B)/(cos A+cos B)+(cos A-cos B)/(sin A+sin B)=0

The value of (sin A+sinB)/(cos A+cos B)+(cos A-cosB)/(sinA-sinB) is……

(sin^2A-sin^2B)/(sinA.cosA-sinB.cosB)=....

cos^(2)(A-B)+cos^(2)B-2cos(A-B).cosA.cosB=?

(b^2-c^2)/(cos B+cosC)+(c^2-a^2)/(cosC+cosA)+(a^2-b^2)/(cosA+cosB)=……

If sinA+sinB=sqrt3(cosB-cosA),then sin3A+sin3B=

Prove that (sinA + sinB)/(cosA + cosB)= tan((A+B)/2)

Prove that: (cosA+cosB)^(2)+(sinA+sinB)^(2)=4cos^(2)(A-B)/2

If sinA+sinB=a and cosA+cosB=b,then cos(A+B)

Prove that: (cosA-cosB)^(2)+(sinA-sinB)^(2)=4sin^(2)(A-B)/(2)

RS AGGARWAL-TRIGONOMETRIC IDENTITIES-Test Yourself
  1. Prove the following identities: tan^2A-tan^2B=(cos^2B-cos^2A)/(cos^2Bc...

    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

    |