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Question : 12
Total: 29
Prove the following :
t a n − 1 √ x
c o s − 1 (
)
OR
Prove the following :
c o s − 1 (
) s i n − 1 (
) s i n − 1 (
)
OR
Prove the following :
Solution:
Let t = t a n − 1 √ x
So√ x
i.e.,t a n 2
On substituting x in the R.H.S. of equationt a n − 1 √ x
c o s − 1 (
)
we get
c o s − 1 (
)
c o s − 1 (
)
Now, using the formula 2θ =c o s − 1 (
)
c o s − 1 (
)
c o s − 1
= t =t a n − 1 √ x
Hence Proved.
OR
Let a be in I quadrant such that
c o s − 1 (
)
So cos a =
⇒ sin a =√ 1 − (
) 2
=√ 1 −
=√
=√
And tan a =
So, a =t a n − 1 (
)
Again b ∊ I quadrant such thats i n − 1 (
)
So, sin b =
⇒ cos b =√ 1 − (
) 2
=√ 1 −
=√
And tan b =
So, b =t a n − 1 (
)
Now, lets i n − 1 (
)
So, sin c =
⇒ cos c =√ 1 − (
) 2
=√ 1 −
=√
√
Ans, tan c =
So c =t a n − 1 (
)
⇒s i n − 1 (
) t a n − 1 (
)
Now, we need t provec o s − 1 (
) s i n − 1 (
) s i n − 1 (
)
Consider a + b
=c o s − 1 (
) + s i n − 1 (
)
=t a n − 1 (
) + t a n − 1 (
)
[c o s − 1 (
) = t a n − 1 (
) s i n − 1 (
) = t a n − 1 (
)
=t a n − 1 (
) t a n − 1 x + t a n − 1 y t a n − 1 (
)
=t a n − 1 (
)
=t a n − 1 (
)
= c =s i n − 1 (
)
Hence Proved.
So
i.e.,
On substituting x in the R.H.S. of equation
we get
Now, using the formula 2θ =
= t =
Hence Proved.
OR
Let a be in I quadrant such that
So cos a =
⇒ sin a =
=
=
=
And tan a =
So, a =
Again b ∊ I quadrant such that
So, sin b =
⇒ cos b =
=
=
And tan b =
So, b =
Now, let
So, sin c =
⇒ cos c =
=
=
Ans, tan c =
So c =
⇒
Now, we need t prove
Consider a + b
=
=
[
=
=
=
= c =
Hence Proved.
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