Question #af5ee

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Answer 1 · 2016-10-14T13:15:08

$x=1/2$

Now
$cos^-1x-sin^-1x=pi/6-------(1)$

using a standard inverse identity ( see proof below)

$cos^-1x+sin^-1x=pi/2-------(2)$

adding eqns. (1) & (2) $sin^-1x$ is eliminated

$2cos^-1x=pi/6+pi/2=(8pi)/12=(2pi)/3$

dividing by 2

$=> cos^-1x=pi/3$

$=>x=1/2$

footnote.

proof of:

$cos^-1x+sin^-1x=pi/2$

Let
$cos^-1x=y$

$=>cosy=x$

$=> sin(pi/2-y)=x$

$=>pi/2-y=sin^-1x$
$=>pi/2=sin^-1x+y$

$=>pi/2=sin^-1x+cos^-1x$