Question

Cosx(tanx+sinxcotx) = sinx+cos^2x

Answer 1

Answer on Question #77468 – Math – Algebra

Question

The equality $\cos x \cdot (\tan x + \sin x \cdot \cot x) = \sin x + (\cos x)^2$ is true for all $x \in R: x \neq \frac{\pi}{2} + \frac{\pi k}{2}, k \in Z$ .

Solution

The function $y = \tan x$ exists for all $x \in R: x \neq \frac{\pi}{2} + \pi k, k \in Z$ .

The function $y = \cot x$ exists for all $x \in R: x \neq \pi k, k \in Z$ .

So consider the expression $\cos x \cdot (\tan x + \sin x \cdot \cot x)$ for all $x \in R: x \neq \frac{\pi}{2} + \frac{\pi k}{2}, k \in Z$ .

$$\begin{array}{l} \cos x \cdot (\tan x + \sin x \cdot \cot x) = \cos x \cdot \left(\frac {\sin x}{\cos x} + \sin x \cdot \frac {\cos x}{\sin x}\right) = \cos x \cdot \left(\frac {\sin x}{\cos x} + \cos x\right) \\ = \sin x + (\cos x) ^ {2} \blacksquare \\ \end{array}$$

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