MHB Finding an Alternative Solution to Doubtful Numerator Steps

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The discussion centers on addressing doubts regarding the derivation of a specific numerator step in a mathematical context. Participants explore whether there are alternative solutions to the problem posed, particularly in relation to the identity $\sin^2x + \cos^2 x=1$. An alternative formulation using complex numbers, $\sin^2x - i^2 \cos^2 x=1$, is suggested for further exploration. The substitution of $x=\frac{2\pi}{9}$ is proposed as a specific case to investigate. Overall, the thread seeks clarity on the validity of these steps and potential solutions.
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Doubt in marked step(Numerator) that how it came from first?
Is there any alternative solution for this?
 
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We generally have $\sin^2x + \cos^2 x=1$.
So $\sin^2x - i^2 \cos^2 x=1$ as well.
Now substitute $x=\frac{2\pi}{9}$. 🤔
 
Thanks
 

Thread 'Area of Overlapping Squares'

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