MHB Solve a trigonometric equation

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The equation to solve is $$\cos^2 \left(\frac{\pi}{4}(\sin x+\sqrt{2}\cos^2 x)\right)-\tan^2 \left(x+\frac{\pi}{4}\tan^2 x\right)=1$$. To approach the solution, moving the tangent function to the other side and applying the Pythagorean Identity is suggested, leading to the conclusion that both terms must equal 1 and 0 respectively. The first term equals 1 when $$\cos^2(\frac{\pi}{4}(\sin x+\sqrt{2}\cos^2 x)) = 1$$, indicating that $$x = 2n\pi - \frac{\pi}{4}$$ is a solution. The second term, $$\tan^2(x+\frac{\pi}{4}\tan^2 x) = 0$$, provides additional solutions including $$x = n\pi$$. The final solutions derived are $$x = 2n\pi - \frac{\pi}{4}$$ and $$x = n\pi$$.
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Solve the equation

$$\cos^2 \left(\frac{\pi}{4}(\sin x+\sqrt{2}\cos^2 x)\right)-\tan^2 \left(x+\frac{\pi}{4}\tan^2 x\right)=1$$
 
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anemone said:
Solve the equation

$$\cos^2 \left(\frac{\pi}{4}(\sin x+\sqrt{2}\cos^2 x)\right)-\tan^2 \left(x+\frac{\pi}{4}\tan^2 x\right)=1$$

I would probably start by moving the tangent function to the other side and applying the Pythagorean Identity to convert everything to cosines...
 
yeah there by it will become product of 2 cosine's =1 that means both equal to 1
 
Last edited:
Prove It said:
I would probably start by moving the tangent function to the other side and applying the Pythagorean Identity to convert everything to cosines...

People posting problems in this sub-forum are presumed to already have worked out the solution in full and are posting the problem as a challenge to others to solve rather than asking for help. (Wink)
 
anemone said:
Solve the equation

$$\cos^2 \left(\frac{\pi}{4}(\sin x+\sqrt{2}\cos^2 x)\right)-\tan^2 \left(x+\frac{\pi}{4}\tan^2 x\right)=1$$

as cos^2 t < = 1 and tan ^2 a >= 0 so the 1st term is 1 and 2nd term is zero.

So cos^2(π/4(sinx+√2 cos2x)) = 1 and tan^2(x+π/4tan^2x) = 0

So (x+π/4tan^2 x) = n π , x = 0 is a solution and other solutions are
= n π – π/4(this I found by guessing tan ^2x =1 and not rigorously)

cos^2(π/4(sin x+√2cos^2x)) = 1 when x = 2 n π – π/4
hence this is the solution
 
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