MHB Solving Trig Equation: Find the Answer Quickly

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The discussion revolves around solving the trigonometric equation $(\sec^4 x +16)^2=2^{12}(4\tan x+1)$. The initial approach involved using the Newton-Raphson method, but a participant suggested substituting $\sec^2(x) = 1 + \tan^2(x)$ to simplify the equation to one in terms of $\tan(x)$. However, this substitution led to a more complex polynomial, ultimately requiring reliance on numerical methods for solutions. The original poster is seeking a more efficient shortcut to solve the problem without complicating it further. The conversation emphasizes the challenge of finding simpler methods for solving complex trigonometric equations.
anemone
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Hi MHB,

Do you think this problem can be approached wisely, rather than expanding it and attack it using the Newton-Raphson method (which I did)?

Problem:

Solve $(\sec^4 x +16)^2=2^{12}(4\tan x+1)$

Thanks for reading and I would appreciate it if in case, you could solve it using shortcut that I failed to acknowledge and share it with me.
 
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anemone said:
Hi MHB,

Do you think this problem can be approached wisely, rather than expanding it and attack it using the Newton-Raphson method (which I did)?

Problem:

Solve $(\sec^4 x +16)^2=2^{12}(4\tan x+1)$

Thanks for reading and I would appreciate it if in case, you could solve it using shortcut that I failed to acknowledge and share it with me.

Well writing $\displaystyle \begin{align*} \sec^2{(x)} = 1 + \tan^2{(x)} \end{align*}$ so that the equation is only in terms of $\displaystyle \begin{align*} \tan{(x)} \end{align*}$ would be a start :)
 
Prove It said:
Well writing $\displaystyle \begin{align*} \sec^2{(x)} = 1 + \tan^2{(x)} \end{align*}$ so that the equation is only in terms of $\displaystyle \begin{align*} \tan{(x)} \end{align*}$ would be a start :)

Thanks, Prove It for your reply. In fact, I solved this problem by using that substitution. I am hoping if you or anyone could find a short cut to approach the problem, since the substitution method led to a more complex polynomial and I at last have to rely wholly on the Newton-Raphson method to find the approximate answers to this problem...(Thinking)
 
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