Question

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Answer 1

Answer:

2 is the correct answer.

Answer 2

$\large\underline{\sf{Given \:Question - }}$

$\sf \: If \: cos\theta = - \: \dfrac{3}{5} \: and \: 180\degree < \theta < 270\degree , \: then \: tan\dfrac{\theta }{2} =$

$\: \: \: \: \: 1) \: \: 2$

$\: \: \: \: \: 2) \: \: - \: 2$

$\: \: \: \: \: 3) \: \: 1$

$\: \: \: \: \: 4) \: \: - \: 1$

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:180\degree < \theta < 270\degree$

and

$\rm :\longmapsto\:cos\theta = - \: \dfrac{3}{5}$

We know that,

$\red{\boxed{ \sf \: cos2x = \frac{1 - {tan}^{2}x}{1 + {tan}^{2} x}}}$

So, using this identity, we get

$\rm :\longmapsto\:\dfrac{1 - {tan}^{2} \dfrac{\theta }{2}}{1 + {tan}^{2} \dfrac{\theta }{2}} = - \: \dfrac{3}{5}$

$\rm :\longmapsto\:3 + 3 {tan}^{2}\dfrac{\theta }{2} = - 5 + 5 {tan}^{2} \dfrac{\theta }{2}$

$\rm :\longmapsto\:3 {tan}^{2}\dfrac{\theta }{2} - 5 {tan}^{2} \dfrac{\theta }{2} = - 5 - 3$

$\rm :\longmapsto\: - \: 2 {tan}^{2}\dfrac{\theta }{2} = -8$

$\rm :\longmapsto\: \: {tan}^{2}\dfrac{\theta }{2} = 4$

$\bf\implies \:tan\dfrac{\theta }{2} = \: \pm \: 2$

But, it is given that

$\rm :\longmapsto\:180\degree < \theta < 270\degree$

$\bf\implies \:90\degree < \dfrac{\theta }{2} < 135\degree$

$\bf\implies \:\dfrac{\theta }{2} \: \in \: {2}^{nd} \: quadrant$

$\bf\implies \:tan\dfrac{\theta }{2} < 0$

$\bf\implies \:tan\dfrac{\theta }{2} \: = \: \: - \: 2$

• Hence, Option 2) is correct.

Additional Information :

$\boxed{ \sf \: sin2x = 2sinx \: cosx}$

$\boxed{ \sf \: cos2x = 1 - {2sin}^{2}x}$

$\boxed{ \sf \: cos2x = {2cos}^{2}x - 1}$

$\boxed{ \sf \: cos2x = {cos}^{2}x - {sin}^{2}x}$

$\boxed{ \sf \: sin2x = \frac{2tanx}{1 + {tan}^{2}x } }$

$\boxed{ \sf \: tan2x = \frac{2tanx}{1 - {tan}^{2}x } }$