Let $${f_k}\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}.x} \right)$$

Consider

$${f_4}\left( x \right) - {f_6}\left( x \right) $$

$$=$$ $${1 \over 4}\left( {{{\sin }^4}x + {{\cos }^4}x} \right) - {1 \over 6}\left( {{{\sin }^6}x + {{\cos }^6}x} \right)$$

$$ = {1 \over 4}\left[ {1 - 2{{\sin }^2}x{{\cos }^2}x} \right] - {1 \over 6}\left[ {1 - 3{{\sin }^2}x{{\cos }^2}x} \right]$$

$$ = {1 \over 4} - {1 \over 6} = {1 \over {12}}$$

From Sine Rule

$${{AB} \over {\sin \theta }} = {{\sqrt {{p^2} + {q^2}} } \over {\sin \left( {\pi - \left( {\theta + \alpha } \right)} \right)}}$$

$$AB = {{\sqrt {{p^2} + {q^2}} \sin \theta } \over {\sin \theta \cos \alpha + \cos \theta \sin \alpha }}$$

$$ = {{\left( {{p^2} + {q^2}} \right)\sin \theta } \over {q\sin \theta + p\cos \theta }}$$

(As $$\cos \alpha = {q \over {\sqrt {{p^2} + {q^2}} }}$$ and $$\sin \alpha = {p \over {\sqrt {{p^2} + {q^2}} }}$$ )

Given expression can be written as

$${{\sin A} \over {\cos A}} \times {{sin\,A} \over {\sin A - \cos A}} + {{\cos A} \over {\sin A}} \times {{\cos A} \over {\cos A - sin\,A}}$$

(As $$\tan A = {{\sin A} \over {\cos A}}$$ and $$\cot A = {{\cos A} \over {\sin A}}$$ )

$$ = {1 \over {\sin A - \cos A}}\left\{ {{{{{\sin }^3}A - {{\cos }^3}A} \over {\cos A\sin A}}} \right\}$$

$$ = {{{{\sin }^2}A + \sin A\cos A + {{\cos }^2}\,A} \over {\sin A\cos A}}$$

$$ = 1 + \sec\, A{\mathop{\rm cosec}\nolimits} \,A$$

Given $$3$$ $$\sin \,P + 4\cos Q = 6$$ $$\,\,\,\,\,\,\,\,...\left( i \right)$$

$$4\sin Q + 3\cos P = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$

Squaring and adding $$(i)$$ & $$(ii)$$ we get

$$9\,{\sin ^2}P + 16{\cos ^2}Q + 24\sin P\cos Q$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 16\,{\sin ^2}Q + 9{\cos ^2}P + 24\sin Q\cos P$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ = 36 + 1 = 37$$

$$ \Rightarrow 9\left( {{{\sin }^2}p + {{\cos }^2}P} \right) + 16\left( {{{\sin }^2}Q + {{\cos }^2}q} \right)$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 24\left( {\sin P\cos Q + \cos P\sin Q} \right) = 37$$

$$ \Rightarrow 9 + 16 + 24\sin \left( {P + Q} \right) = 37$$

[ As $${\sin ^2}\theta + {\cos ^2}\theta = 1$$ and

$$\sin A\cos B + \cos A\sin B$$ $$ = \sin \left( {A + B} \right)$$ ]

$$ \Rightarrow \sin \left( {P + Q} \right) = {1 \over 2}$$

$$ \Rightarrow P + Q = {\pi \over 6}$$ or $${{5\pi } \over 6}$$

$$ \Rightarrow R = {{5\pi } \over 6}$$ or $${\pi \over 6}$$

(as $$P + Q + R = \pi $$ )

If $$R = {{5\pi } \over 6}$$ then $$0 < P,Q < {\pi \over 6}$$

$$ \Rightarrow \cos Q < 1$$ and $$\sin P < {1 \over 2}$$

$$ \Rightarrow 3\sin P + 4\cos Q < {{11} \over 2}$$ which is not true.

So $$R = {\pi \over 6}$$