If P:Q = Tan 2A :Cos A and Q:R = Cos2A:Sin2A then P:R =?

1) Tan2A 2) 2 SinA 3)1 4)SecA

Answer :

We have P : Q  = Tan 2A  :  Cos A , So

$\frac{P}{Q} = \frac{T\mathrm{an} 2A}{Cos A}$  ------------------------- ( 1 )

And

Q : R  = Cos 2A  :  Sin 2A , So

$\frac{Q}{R} = \frac{Cos 2A}{Sin2 A}$ 

$\frac{Q}{R} = Cot 2A$  ( As  we know $\frac{Cos \theta }{Sin \theta }$  =  Cot $\theta$ )

And

$\frac{Q}{R} = \frac{1}{Tan 2A}$                                       ----------------------- ( 2 ) ( As  we know  Cot $\theta$ = $\frac{1}{Tan \theta }$)

Now we multiply equation 1 and 2 , we get

$$\begin{gathered} \Rightarrow \frac{P}{Q} \times \frac{Q}{R}= \frac{T\mathrm{an} 2A}{Cos A} \times \frac{1}{Tan 2A} \\ \Rightarrow \frac{P}{R} = \frac{1}{Cos A} \\ \Rightarrow \frac{P}{R} = Sec A ( As we know \frac{1}{Cos \theta } = Sec \theta ) \end{gathered}$$ 
So,

P :  R  =  Sec A

Option ( 4 )  ( Ans )