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((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cot...

`((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A`

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We have
`((1+tan^(2) A)/(1+ cot^(2) A))=(sec^(2) A)/("cosec"^(2)A)`
` [ because 1+ tan^(2) A= sec^(2) A " and " 1+ cot^(2) A = " cosec"^(2)A] `
` = (1)/(cos^(2) A)* sin^(2) A= (sin^(2) A)/(cos^(2) A)= tan^(2) A. `
And, `((1- tan A)/(1- cot A))^(2)= ((1-(sin A)/(cos A))^(2))/((1- (cos A)/(sin A))^(2))`
` =(((cos A - sin A)^(2))/(cos^(2)A))/(((sin A - cos A)^(2))/(sin^(2)A))= (sin^(2)A)/(cos^(2)A)= tan^(2) A.`
` therefore ((1+ tan^(2)A)/(1+ cot^(2) A))=((1- tan A)/(1- cot A))^(2) = tan ^(2) A. `
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RS AGGARWAL-TRIGONOMETRIC IDENTITIES-Test Yourself
  1. ((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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