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Prove that : tan^-1(1/4)+tan^-1(2/9)=1/2...

Prove that : `tan^-1(1/4)+tan^-1(2/9)=1/2 cos^-1(3/5)`.

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LHS `=tan^(-1)1/4+tan^(-1)2/9`, where `xy=(1/4 xx 2/9)=1/18 lt 1`
`=tan^(-1){(1/4+2/9)/(1-1/4 xx 2/9)}=tan^(-1)(17/34)=tan^(-1)1/2`.
Now, let `1/2cos^(-1)3/5=theta`. Then, `cos2theta=3/5`.
`therefore tantheta = sqrt((1-cos2theta)/(1+cos2theta))=sqrt((1-3/5)/(1+3/5))=sqrt(1/4)=1/2`
`rArr tan^(-1)1/2=theta rArr tan^(-1)1/2=1/2cos^(-1)3/5`.
Hence, `tan^(-1)1/4+tan^(-1)2/9=1/2cos^(-1)3/5`.
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RS AGGARWAL-INVERSE TRIGNOMETRIC FUNCTIONS -Objective Questons
  1. Prove that : tan^-1(1/4)+tan^-1(2/9)=1/2 cos^-1(3/5).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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