Question

if cos inverse x minus cos inverse x =0 write down the value of x​

Answer 1

Proof:

Let, cos−1 x = α and cos−1 y = β

From cos−1 x = α we get,

x = cos α

and from cos−1 y = β we get,

y = cos β

Now, cos (α - β) = cos α cos β + sin α sin β

⇒ cos (α - β) = cos α cos β + 1−cos2α−−−−−−−−√ 1−cos2β−−−−−−−−√

⇒ cos (α - β) = (xy + 1−x2−−−−−√1−y2−−−−−√)

⇒ α - β = cos−1(xy + 1−x2−−−−−√1−y2−−−−−√)

or, cos−1 x - cos−1 y = cos−1(xy + 1−x2−−−−−√1−y2−−−−−√)

Therefore, arccos(x) - arccos(y) = arccos(xy) + 1−x2−−−−−√1−y2−−−−−√) Proved.

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