Question

if tan theta equals to 1 then find the values of sin theta + cos theta upon sec theta + cosec theta​

Answer 1

ɢɪᴠᴇɴ:-

•  $\tan\:\theta\:=\:\sf\:1$

ᴛᴏ ꜰɪɴᴅ:-

• $\sf\:\dfrac{\sin\:\theta\:+\:\cos\:\theta}{\sec\:\theta\:+\:\csc\:\theta}$

From trigonometry table,

we know,

$\begin{lgathered}\tan\:\sf\:45^{\circ}\:=\:1\\\\\\\therefore\sf\:\theta\:=\:45^{\circ}\end{lgathered}$

Then,

$\begin{lgathered}\sf\:\dfrac{\sin\:\theta\:+\:\cos\:\theta}{\sec\:\theta\:+\:\cosec\:\theta}\\\\\\\sf\:Now\:,\\\\\\:\implies\sf\:\sin\:\theta\:=\:\sin\:45^{\circ}\:=\:\dfrac{1}{\sqrt{2}}\\\\\\:\implies\sf\:\cos\:\theta\:=\:\cos\:45^{\circ}\:=\:\dfrac{1}{\sqrt{2}}\\\\\\:\implies\sf\:\sec\:\theta\:=\:\sec\:45^{\circ}\:=\:\sqrt{2}\\\\\\:\implies\sf\:\csc\:\theta\:=\:\csc\:45^{\circ}\:=\:\sqrt{2}\end{lgathered}$

Now,

$\begin{lgathered}\sf\:\dfrac{\sin\:\theta\:+\:\cos\:\theta}{\sec\:\theta\:+\:\csc\:\theta}\\\\\\\sf\:=\:\dfrac{\sin\:45^{\circ}\:+\:\cos\:45^{\circ}}{\sec\:45^{\circ}\:+\:\csc\:45^{\circ}}\\\\\\\displaystyle\sf\:=\:\dfrac{\dfrac{1}{\sqrt{2}}\:+\:\dfrac{1}{\sqrt{2}}}{\sqrt{2}\:+\:\sqrt{2}}\\\\\\\displaystyle\sf\:=\:\dfrac{\dfrac{1\:+\:1}{\sqrt{2}}}{2\:\sqrt{2}}\\\\\\\displaystyle\sf\:=\:\dfrac{\dfrac{2}{\sqrt{2}}}{2\:\sqrt{2}}\end{lgathered}$

$\begin{lgathered}\displaystyle\sf\:=\:\dfrac{2}{\sqrt{2}}\:\div\:2\:\sqrt{2}\\\\\\\sf\:=\:\dfrac{\cancel2}{\sqrt{2}}\:\times\:\dfrac{1}{\cancel{2}\:\sqrt{2}}\\\\\\\sf\:=\:\dfrac{1}{\sqrt{2}\:\times\:\sqrt{2}}\\\\\\\sf\:=\:\dfrac{1}{(\:\sqrt{2}\:)^2}\\\\\\\sf\:=\:\dfrac{1}{2}\\\\\\\end{lgathered}$

$:\implies\bf\dfrac{\sin\:\theta\:+\:\cos\:\theta}{\sec\:\theta\:+\:\csc\:\theta}\:=\:\dfrac{1}{2}$

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$\large{\pink{\underline{\tt{Let's\:know\:more!♡}}}}$

$\Large{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 65^{\circ} & \sf 90^{\circ} \\ \cline{1-6} \sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} & 1 \\ \cline{1-6} \cos & 1 & \dfrac{ \sqrt{ 3 }}{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \cline{1-6} \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \cline{1-6} \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \cline{1 - 6} \sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \cline{1-6} \csc & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \cline{1 - 6}\end{tabular}}$

Answer 2

Answer:

The Above Answer Is Correct ......