secθ-cosθ=(cotθ+cosθ)(tanθ-sinθ)

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on Jan 20, 2025 at 5:55 am



You asked:

Solve the equation \(\sec\left( θ \right) - \cos\left( θ \right) = \left(\cot\left( θ \right) + \cos\left( θ \right)\right) \left(\tan\left( θ \right) - \sin\left( θ \right)\right)\) for the variable \(θ\).

MathBot Answer:

The real solutions are: \[\left\{2 n \pi\; \middle|\; n \in \mathbb{Z}\right\}\]

The complex solutions are: \[\begin{aligned}θ &= \left\{\frac{\pi}{4} + 2 \pi n - \frac{i \log{\left(2 + \sqrt{3} \right)}}{2}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 0.78539816 - 0.65847895 i\; \middle|\; n \in \mathbb{Z}\right\}\\θ &= \left\{\frac{5 \pi}{4} + 2 \pi n - \frac{i \log{\left(2 + \sqrt{3} \right)}}{2}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 3.9269908 - 0.65847895 i\; \middle|\; n \in \mathbb{Z}\right\}\\θ &= \left\{\frac{\pi}{4} + 2 \pi n - \frac{i \log{\left(2 - \sqrt{3} \right)}}{2}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 0.78539816 + 0.65847895 i\; \middle|\; n \in \mathbb{Z}\right\}\\θ &= \left\{\frac{5 \pi}{4} + 2 \pi n - \frac{i \log{\left(2 - \sqrt{3} \right)}}{2}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 3.9269908 + 0.65847895 i\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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