Question

if tan square theta - 3 sec square theta + 3 = 0 , then what is the value of sin theta + cos theta​

Answer 1

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

Given that

$\rm :\longmapsto\:0\degree < \theta < 90\degree$

and

$\rm :\longmapsto\: {tan}^{2}\theta - 3sec\theta + 3 = 0$

We know,

$\boxed{ \bf{ {sec}^{2}\theta - {tan}^{2} \theta = 1}}$

So, using this

$\rm :\longmapsto\: {sec}^{2}\theta - 1 - 3sec\theta + 3 = 0$

$\rm :\longmapsto\: {sec}^{2}\theta - 3sec\theta + 2= 0$

$\rm :\longmapsto\: {sec}^{2}\theta - 2sec\theta - sec\theta + 2= 0$

$\rm :\longmapsto\:sec\theta (sec\theta - 2) - 1(sec\theta - 2) = 0$

$\rm :\longmapsto\:(sec\theta - 1)(sec\theta - 2) = 0$

$\rm :\implies\:sec\theta = 1 \: \: \rm \implies\:\theta = 0\degree \: \{rejected \}$

or

$\rm \implies\:sec\theta = 2 \: \: \rm \implies\:\theta = 60\degree$

Now,

Consider,

$\rm :\longmapsto\:sin\theta + cot\theta$

$\rm \: \: = \:sin60\degree + cot60\degree$

$\rm \: \: = \:\dfrac{ \sqrt{3} }{2} + \dfrac{1}{ \sqrt{3} }$

$\rm \: \: = \:\dfrac{ \sqrt{3} }{2} + \dfrac{1}{ \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} }$

$\rm \: \: = \:\dfrac{ \sqrt{3} }{2} + \dfrac{ \sqrt{3} }{ 3 }$

$\rm \: \: = \:\dfrac{3 \sqrt{3} + 2 \sqrt{3} }{6}$

$\rm \: \: = \:\dfrac{5 \sqrt{3}}{6}$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{\boxed{\bf{ sin\theta + cot\theta \rm \: \: = \:\dfrac{5 \sqrt{3}}{6}}}}$

So,

Option (a) is correct.

Additional Information :-

$\boxed{ \bf{ {sin}^{2} x + {cos}^{2} x = 1}}$

$\boxed{ \bf{ {sec}^{2} x - {tan}^{2} x = 1}}$

$\boxed{ \bf{ {cosec}^{2} x - {cot}^{2} x = 1}}$

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$