MHB Solving Trig Equation: Find the Answer Quickly

[anemone]

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.

 

[Prove It]

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 :)

 

[anemone]

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)