\( \sin ^{5} \theta+\cos ^{6} \theta=\frac{1}{4}\left(1+3 \cos ^{2} 2 \theta\right) \) (b) \( \cos 4 \theta=1-8 \sin ^{2} \theta+8 \sin ^{4} \theta \) LHS \( =\sin ^{6} \theta+\cos ^{6} \theta \)
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Did you know that the relationship between sine and cosine functions has been explored for centuries? Mathematicians like Pythagoras and later Euler helped develop the foundational principles of trigonometry, which are essential in solving equations such as the one you're working with. By expressing sine and cosine in various powers, they were able to uncover deeper connections in both geometry and algebra! Now, when you encounter equations involving \(\sin\) and \(\cos\) raised to different powers, a common trick is to exploit identities like \(\sin^2 \theta + \cos^2 \theta = 1\) or even the double angle formulas. A classic mistake is to forget these identities, leading to overly complicated expressions. So, keep those identities handy and simplify wisely before diving into complex calculations!
