Question

if cosec alpha +cot alpha =2√3 than prove that cos alpha =2\√5

Answer 1

Answer on the Question #80408 – Math – Trigonometry

Question

if cosec alpha +cot alpha = 2√3 than prove that cos alpha = 2√5

Solution

Now, let’s solve the following equation:

$$\begin{array}{l} \operatorname{Cosec}(x) + \cot(x) = 2 \cdot \sqrt{3}; \\ (1 + \cos(x)) / \sin(x) = 2 \cdot \sqrt{3}; \quad | \wedge 2 \\ (1 + \cos(x))^2 = 12 \cdot \sin^2(x); \\ (1 + \cos(x))^2 = 12 \cdot (1 - \cos^2(x)); \\ 1 + 2 \cdot \cos(x) + \cos^2(x) = 12 - 12 \cdot \cos^2(x); \\ 13 \cdot \cos^2(x) + 2 \cdot \cos(x) - 11 = 0; \\ D = 4 + 4 \cdot 13 \cdot 11 = 576 = 24^2; \\ \cos(x) = (-2 \pm 24) / 26; \\ \cos(x)_1 = -1; \\ \cos(x)_2 = 22/26 = 11/13; \end{array}$$

So, $\cos(x)$ can be $-1$ or $11/13$, it is not equal to $2\sqrt{5}$, hence the statement of question is false.

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