If $\cos \left(2 \theta+54^{\circ}\right)=\sin \theta, 0^{\circ}<\left(2 \theta+54^{\circ}\right)<90^{\circ}$, then what is the value of $\frac{1}{\cot 5 \theta+\sec \frac{5 \theta}{2}} ?$

$\frac{\sqrt{3}}{3}$

We are given :-

cos (2θ + 54º) = sinθ

{ using, Iff A + B = 90º , then sinA = cosB }

So, 2θ + 54º + θ= 90º

3θ = 36º 

θ = 12º

Now,

\(\frac{1}{ cot5θ + sec5θ/2}\)

= \(\frac{1}{ cot60º + sec30º }\)

= \(\frac{1}{ 1/√3 + 2/√3 }\)

= \(\frac{√3}{ 3 }\)