Question

If `sin^(-1)a+sin^(-1)b+sin^(-1)c=pi,` then `asqrt(1-a^2)+bsqrt(1-b^2)+csqrt(1-c^2)` is equal to `a+b+c` (b) `a^2b^2c^2` `2a b c` (d) `4a b c`

Answer 1

Let `sin^-1a = x,sin^-1b = y and sin^-1c = z`
Then, `a = sinx, b = siny, c = sinz`
and `x+y+z = pi`
Now, `asqrt(1-a^2)+bsqrt(1-b^2)+csqrt(1-c^2)`
`=sinxsqrt(1-sin^2x)+sinysqrt(1-sin^2y)+sinzsqrt(1-sin^2z)`
`=sinxcosx+sinycosy+sinzcosz`
`=1/2(sin2x+sin2y+sin2z)`
As `x+y+z =pi `,
`:. sin2x+sin2y+sin2z = 4sinxsinysinz`
`:. 1/2(sin2x+sin2y+sin2z) = 1/2(4sinxsinysinz)`
`=2(sinxsinysinz)`
`=2abc`
`:. asqrt(1-a^2)+bsqrt(1-b^2)+csqrt(1-c^2) = 2abc`
So, option - `(c)` is the correct option.