Impossible Trig! Need Immediate Help

[DriftGhost]

Simplify and solve for x:

(tanx)^2=(sqrt(2)/2)(secx)

 

[Gene]

Usually it is best to express things in sines and cosines.
(tan(x))^2=(sqrt(2)/2)(sec(x))
sin²(x)/cos²(x)=(sqrt(2)/2)(1/cos(x)
sin²(x)=(sqrt(2)/2)*cos(x))
(1-cos²(x))=(sqrt(2)/2)*cos(x)
cos²(x)+(sqrt(2)/2)cos(x)-1 = 0
A quadratic in cos(x)

 

[soroban]

Hello, DriftGhost!

Or you can use a secant-tangent identity:

We have: \(\displaystyle \;\underbrace{\tan^2x}\;=\;\frac{\sqrt{2}}{2}\cdot\sec x\)

Then: \(\displaystyle \;\overbrace{\sec^2x\,-\,1}\;=\;\frac{\sqrt{2}}{2}\cdot\sec x\)

Multiply by 2: \(\displaystyle \;2\cdot\sec^2x\,-\,2\;=\;\sqrt{2}\cdot\sec x\)

And we have the quadratic: \(\displaystyle \;2\cdot\sec^2x \,-\,\sqrt{2}\cdot\sec x\,-\,2\;=\;0\)

Quadratic Formula: \(\displaystyle \:\sec x \;= \;\frac{-(-\sqrt{2})\,\pm\,\sqrt{(-\sqrt{2})^2\,-\,4(2)(-2)}}{2(2)}\;=\;\frac{\sqrt{2}\,\pm\,2\sqrt{5}}{4}\)

We have: \(\displaystyle \,\sec x\:=\:1.47158379\;\;\Rightarrow\;\;x\:\approx\:47.2^o\:=\:0.824\) radians

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We also have: \(\displaystyle \:\sec x\:=\:-0.764480598\)

But since \(\displaystyle \,|\sec x|\:\geq\:1\), there is no second solution.