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# Trigonometry Identities Help

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how do you expand, then express in a single trig ratio and/or constant?

4sinxcosx (sec^3x cosx + cscxsecx)

Guest Nov 29, 2018

#1
+20633
+5

how do you expand, then express in a single trig ratio and/or constant?
$$4 \sin(x) \cos(x) \Big(\sec^3(x) \cos(x) + \csc(x) \sec(x)\Big)$$

$$\text{Formula:}\\ \text{\csc(x) = \dfrac{1}{\sin(x)} } \\ \text{\sec(x) = \dfrac{1}{\cos(x)} } \\$$

$$\begin{array}{|rcll|} \hline && 4 \cdot\sin(x) \cos(x) \Big(\sec^3(x) \cos(x) + \csc(x) \sec(x)\Big) \\\\ &=& 4 \cdot\sin(x) \cos(x) \left( \dfrac{\cos(x)}{\cos^3(x)} + \dfrac{1}{\cos(x)\sin(x)} \right) \\\\ &=& 4 \cdot\sin(x) \cos(x) \left( \dfrac{1}{\cos^2(x)} + \dfrac{1}{\cos(x)\sin(x)} \right) \\\\ &=& 4 \cdot\left( \dfrac{\sin(x) \cos(x)}{\cos^2(x)} + \dfrac{\sin(x) \cos(x)}{\cos(x)\sin(x)} \right) \\\\ &=& 4 \cdot\left( \dfrac{\sin(x)}{\cos(x)} + 1 \right) \quad | \quad \dfrac{\sin(x)}{\cos(x)}=\tan(x) \\\\ &\mathbf{=}& \mathbf{ 4 \cdot \Big( \tan(x) + 1 \Big )} \\ \hline \end{array}$$

heureka  Nov 29, 2018
#1
+20633
+5

how do you expand, then express in a single trig ratio and/or constant?
$$4 \sin(x) \cos(x) \Big(\sec^3(x) \cos(x) + \csc(x) \sec(x)\Big)$$

$$\text{Formula:}\\ \text{\csc(x) = \dfrac{1}{\sin(x)} } \\ \text{\sec(x) = \dfrac{1}{\cos(x)} } \\$$

$$\begin{array}{|rcll|} \hline && 4 \cdot\sin(x) \cos(x) \Big(\sec^3(x) \cos(x) + \csc(x) \sec(x)\Big) \\\\ &=& 4 \cdot\sin(x) \cos(x) \left( \dfrac{\cos(x)}{\cos^3(x)} + \dfrac{1}{\cos(x)\sin(x)} \right) \\\\ &=& 4 \cdot\sin(x) \cos(x) \left( \dfrac{1}{\cos^2(x)} + \dfrac{1}{\cos(x)\sin(x)} \right) \\\\ &=& 4 \cdot\left( \dfrac{\sin(x) \cos(x)}{\cos^2(x)} + \dfrac{\sin(x) \cos(x)}{\cos(x)\sin(x)} \right) \\\\ &=& 4 \cdot\left( \dfrac{\sin(x)}{\cos(x)} + 1 \right) \quad | \quad \dfrac{\sin(x)}{\cos(x)}=\tan(x) \\\\ &\mathbf{=}& \mathbf{ 4 \cdot \Big( \tan(x) + 1 \Big )} \\ \hline \end{array}$$

heureka  Nov 29, 2018