perplexus.info :: Just Math : cosine equated to polynomial

cosine equated to polynomial (Posted on 2024-02-13)

Find all real x and y such that cos²x = (1 + y)²/(4y)

No Solution Yet Submitted by Danish Ahmed Khan

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Puzzle Answer

| Comment 1 of 3

sin^2x = 1-(1 + y)^2/(4y)= (1-y)^2/(4y)

=> tan^2(x) = (1-y)^2/(1+y)^2

=> tan(x) = (1-y)/(1+y)

= {tan (2n*pi+pi/4) - tan(z)}/{1+ tan(2n*pi+pi/4)*tan(z)}, where y = tanz

= tan (2n*pi+pi/4 - z)

Therefore:

x+z = (pi/4)*(8n+1)

= > x+ arctan(y) = (pi/4)*(8n+1)

Posted by K Sengupta on 2024-02-13 07:11:12

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