How to Find Maximum and Minimum Values of Sin^4theta + Cos^4theta

AI Thread Summary
The discussion focuses on proving the identity sin^4θ + cos^4θ = 1/4(3 + cos4θ) and finding its maximum and minimum values. Participants suggest starting from the right-hand side (RHS) and manipulating the expression to simplify it. One user successfully rearranges the terms to facilitate factoring, leading to progress in the proof. The conversation highlights the importance of strategic algebraic manipulation in solving trigonometric identities. Overall, the thread emphasizes collaborative problem-solving in mathematics.
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Homework Statement



prove that sin^4theta + cos^4 theta =1/4 (3+cos4theta).hence,find the greatest and least value of sin^4theta + cos^4 theta.

please give hints to start . no idea at all...i start from the RHS and get 2cos^4 theta - 2 cos^2 theta +1 , then how sin^4theta + cos^4 theta



Homework Equations





The Attempt at a Solution

 
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hibernator said:

Homework Statement



prove that sin^4theta + cos^4 theta =1/4 (3+cos4theta).hence,find the greatest and least value of sin^4theta + cos^4 theta.

please give hints to start . no idea at all...i start from the RHS and get 2cos^4 theta - 2 cos^2 theta +1 , then how sin^4theta + cos^4 theta

So you started at the RHS and got to this:
2cos4 θ - 2cos2 θ + 1.
You're really close, then. Split the 2cos4 θ and rearrange the terms like so:
cos4 θ + 1 - 2cos2 θ + cos4 θ
Factor the part in bold.
 
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Amazing idea that i never thought before ! TQ so much eumyang..
 

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