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if, sin^-1x + sin^-1y + sin^-1z =pi then...

if, `sin^-1x + sin^-1y + sin^-1z =pi` then prove that `xsqrt(1-x^2) + ysqrt(1-y^2) + zsqrt(1-z^2) =2xyz.`

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Verified by Experts

Let
`sin^(−1)x=A,sin^(−1)y=B,sin^(−1)z=c`
`x=sinA,y=sinB,z=sinc`
`=>A+B+C=π`
`=>sin2A+sin2B+sin2C=9sinAsinBsinC`
LHS
`=2sin(A+B)⋅cos(A−B)+sin2C`
`=2sin(π−C)⋅cos(A−B)+2sinCcosC`
`=2sinC(cos(A−B)+cosC)`
...
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RD SHARMA-INVERSE TRIGONOMETRIC FUNCTION-Solved Examples And Exercises
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  2. If tan^(-1){(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))}=alpha...

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  5. Evaluate: sin{tan^(-1)(7/(24))}

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  7. Solve : "cos"(sin^(-1)x)=1/6

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  11. If cj >0 for i=1,2,....., n prove that tan^(-1)((c1x-y)/(c1y+x)...

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  13. Prove that: sum(m=1)^ntan^(-1)((2m)/(m^4+m^2+2))=tan^(-1)((n^2+n)/(n^2...

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  14. Sum the following series to infinity: tan^(-1)1/(1+1+1^2)+tan^(-1)1/(...

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  15. Solve the equation: tan^(-1)sqrt(x^2+x)+sin^(-1)sqrt(x^2+x+1)=pi/2

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  16. If a1,a2,a3, an are in arithmetic progression with common difference d...

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  17. Prove that: tan^(-1)((1-x)/(1+x))-tan^(-1)((1-y)/(1+y))=sin^(-1)((y-x...

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  18. If x > y > z >0, then find the value of cot^(-1)(x y+1)/(x-y)+cot^(-1)...

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  19. Find the maximum and minimum values of (sin^(-1)x)^3+(cos^(-1)x)^3, wh...

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  20. the greatest and least values of (sin^(-1) x)^2+(cos^(-1)x)^2

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