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Question
Prove the following:
`cos^-1(3/5) + cos^-1(4/5) = pi/(2)`
Solution
Let `cos^-1(3/5)` = x
∴ cos x = `(3)/(5), "where" 0 < x < pi/(2)`
∴ sin x > 0
Now,
sin x = `sqrt(1 - cos^2x)`
= `sqrt(1 - 9/25)`
= `sqrt(16/25)`
= `(4)/(5)`
∴ x = `sin^-1(4/5)`
∴ `cos^-1(3/5) = sin^-1(4/5)` ...(1)
L.H.S. = `cos^-1(3/5) + cos^-1(4/5)`
= `sin^-1(4/5) + cos^-1(4/5)` ...[By (1)]
= `pi/(2) ...[∵ sin^-1x + cos^-1x = pi/2]`
= R.H.S.
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