Question

if theta is an acute angle and tan theta +cot theta =2 then find the value of tan^25 theta+cot^25theta+sec^2 theta+ cos^2theta​

Answer 1

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

Given Trigonometric equation is

$\rm :\longmapsto\:tan\theta + cot\theta = 2$

We know,

$\boxed{ \bf{ \:cotx = \frac{1}{tanx}}}$

So, using this, we get

$\rm :\longmapsto\:tan\theta + \dfrac{1}{tan\theta } = 2$

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

$\rm :\longmapsto\: {tan}^{2}\theta + 1 = 2tan\theta$

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

We know,

$\boxed{ \bf{ \: {x}^{2} + {y}^{2} - 2xy = {(x - y)}^{2}}}$

So, using this, we get

$\rm :\longmapsto\: {(tan\theta - 1)}^{2} = 0$

$\rm :\longmapsto\:tan\theta - 1 = 0$

$\rm :\longmapsto\:tan\theta = 1$

$\rm :\longmapsto\:tan\theta = tan45 \degree$

$\bf\implies \:\boxed{ \bf{ \:\theta = 45\degree}}$

Now, Consider,

$\rm :\longmapsto\: {tan}^{25}\theta + {cot}^{25}\theta + {sec}^{2}\theta + {cos}^{2}\theta$

On substituting the value, we get

$\rm \:=\: {tan}^{25}45\degree + {cot}^{25}45\degree + {sec}^{2}45\degree + {cos}^{2}45\degree$

We know,

$\boxed{ \bf{ \:tan45\degree = 1}} \: \: \boxed{ \bf{ \:cot45\degree = 1}} \: \: \\ \\ \boxed{ \bf{ \:sec45\degree = \sqrt{2}}} \: \: \boxed{ \bf{ \:cos45\degree = \frac{1}{ \sqrt{2} }}}$

So, on substituting these values, we get

$\rm \:  =  \: {1}^{25} + {1}^{25} + {( \sqrt{2}) }^{2} + {\bigg[\dfrac{1}{ \sqrt{2} } \bigg]}^{2}$

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

$\rm \:  =  \:4 + \dfrac{1}{2}$

$\rm \:  =  \: \dfrac{8 + 1}{2}$

$\rm \:  =  \: \dfrac{9}{2}$

Hence,

$\rm :\longmapsto\: \boxed{ \bf{ \:{tan}^{25}\theta + {cot}^{25}\theta + {sec}^{2}\theta + {cos}^{2}\theta = \frac{9}{2}}}$

Additional Information :-

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