If P, Q and R are angles of an isosceles tria | vedclass

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(B) Given that $P, Q$ and $R$ are angles of an isosceles triangle and $\angle P = \frac{\pi}{2}$.Since $P + Q + R = \pi$, we have $Q + R = \pi - \frac

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$-i$

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(B) Given that $P, Q$ and $R$ are angles of an isosceles triangle and $\angle P = \frac{\pi}{2}$.Since $P + Q + R = \pi$, we have $Q + R = \pi - \frac{\pi}{2} = \frac{\pi}{2}$.Since the triangle is isosceles and $\angle P = \frac{\pi}{2}$, we have $Q = R = \frac{\pi}{4}$.Using the identity $\cos 	heta + i \sin 	heta = e^{i 	heta}$, the expression becomes:$E = (e^{-i P/3})^3 + (e^{i Q})(e^{-i R}) + (e^{-i P})(e^{-i Q})(e^{-i R})$$E = e^{-i P} + e^{i(Q - R)} + e^{-i(P + Q + R)}$Substituting $P = \frac{\pi}{2}, Q = \frac{\pi}{4}, R = \frac{\pi}{4}$:$E = e^{-i \pi/2} + e^{i(0)} + e^{-i(\pi/2 + \pi/4 + \pi/4)}$$E = -i + 1 + e^{-i \pi}$$E = -i + 1 - 1 = -i$.

If $P, Q$ and $R$ are angles of an isosceles triangle and $\angle P = \frac{\pi}{2}$, then the value of $\left(\cos \frac{P}{3} - i \sin \frac{P}{3}\right)^3 + (\cos Q + i \sin Q) (\cos R - i \sin R) + (\cos P - i \sin P) (\cos Q - i \sin Q) (\cos R - i \sin R)$ is:

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