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Prove that: tan^(-1)(4/5)+cos^(-1)(12/13...

Prove that: `tan^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

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Verified by Experts

We have,
LHS `=cos^(-1)4/5+cos^(-1)12/13`
`=cos^(-1){(4/5 xx 12/13)-sqrt(1-(4/5)^(2)).sqrt(1-(12/13)^(2))}`
`=cos^(-1){48/65-sqrt(1-(16/25)).sqrt(1-144/169)}`
`=cos^(-1){48/65-sqrt(9/25).sqrt(25/169)}`
`=cos^(-1){48/65-3/5 xx 5/13}=cos^(-1)(48/65-15/65)`
`=cos^(-1){12/17-sqrt(64/289).sqrt(9/25)}=cos^(-1){12/17 -8/17 xx 3/5)`
`=cos^(-1){12/17-24/85}=cos^(-1)(36/85)=`RHS
Hence, `sin^(-1)8/17+sin^(-1)3/5=cos^(-1)36/85`.
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RS AGGARWAL-INVERSE TRIGNOMETRIC FUNCTIONS -Objective Questons
  1. Prove that: tan^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)

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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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