Put x=cos‌2‌θ
⇒y=sin‌−1
(‌

√1+cos‌2‌θ

2

+‌

√1−cos‌2‌θ

2

)
⇒y=sin‌−1
(‌

√2cos22θ

2

+‌

√2⋅sin‌2θ

2

)
⇒y=sin‌−1(‌

cos‌2‌θ

√2

+‌

sin‌2θ

√2

)
⇒y=sin‌−1(sin‌(‌

π

4

+2θ).
⇒y=‌

π

4

+2θ‌. ‌
⇒‌

dy

dθ

=2
‌ Put ‌θ=‌

cos−1x

2

⇒‌

dθ

dx

=‌

−1

4√1−x2

∴‌

dy

dx

=‌

−1

2√1−x2

OR

As we know that exponential and cosine functions are continuous and differentiable on R .
Let us find the values of the function at an extreme
⇒f(−‌

π

2

)=e−‌

π

2

‌cos(−‌

π

2

)
⇒f(−‌

π

2

)=e−‌

π

2

×0
⇒f(−‌

π

2

)=0
⇒f(‌

π

2

)=e‌

π

2

‌cos(‌

π

2

)
⇒f(π)=e‌

π

2

×0
⇒f(π)=0
Here, f′(−π∕2)=f(π∕2) , therefore there exist a c∈(−π∕2,π∕2) such that f′(c)=0 .
Let us find the derivative of f(x)
⇒f′(x)=‌

d(ex‌cos‌x)

dx

⇒f′(x)=cos‌x‌

d(ex)

dx

+ex‌

d(cos‌x)

dx

⇒f′(x)=ex(−sin‌x+cos‌x)
‌ Here, ‌f′(c)=0
⇒ec(−sin‌c+cos‌c)=0
⇒−sin‌c+cos‌c=0
⇒‌

−1

√2

sin‌c+‌

1

√2

‌cos‌c=0
⇒−sin‌(‌

π

4

) sinc+cos(‌

π

4

)‌cos‌c=0
⇒cos(c+‌

π

4

)=0
⇒c+‌

π

4

=‌

π

2

⇒c=‌

π

4

E(−‌

π

2

,‌

π

2

)
Thus, Rolle's theorem is verified.