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  • Find the range of function  <img alt="\tan \frac{\alpha}{2} \cdot \tan \alpha" src="https://learn.careers360.com/latex-image/?%5Ctan%20%5Cfrac%7B%5Calpha%7D%7B2%7D%20%5Ccdot%20%5Ctan%20%5

Find the range of function  \tan \frac{\alpha}{2} \cdot \tan \alpha

Option: 1

(-\infty,-2] U[0,\infty)


Option: 2

(-\infty,\infty)


Option: 3

(-\infty,-1] U[1,\infty)


Option: 4

(-\infty,-1] U[0,\infty)


Answers (1)

best_answer

Half Angle Formula -

Half Angle Formula

 

 

\begin{aligned}1.\;\;\; \sin \left(\frac{\alpha}{2}\right) &=\pm \sqrt{\frac{1-\cos \alpha}{2}} \\ 2.\;\;\;\cos \left(\frac{\alpha}{2}\right) &=\pm \sqrt{\frac{1+\cos \alpha}{2}} \\3.\;\;\; \tan \left(\frac{\alpha}{2}\right) &=\pm \sqrt{\frac{1-\cos \alpha}{1+\cos \alpha}} =\frac{\sin \alpha}{1+\cos \alpha}=\frac{1-\cos \alpha}{\sin \alpha} \end{aligned}

 

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\tan \frac{\alpha}{2} \cdot \tan \alpha=(\frac{1-\cos \alpha}{\sin \alpha}) \cdot \frac{\sin \alpha}{\cos \alpha}\\ =\sec \alpha -1\\ \text{ Range of sec } \alpha (-\infty,-1] U[1,\infty)\\ \text{ Range of } f(\alpha) is (-\infty,-2] U[0,\infty)\\

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Riya

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