Question

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

Answer 1

$tan \theta \: = 1 = tan45 \degree \\ \therefore \: \fbox{\theta = 45 \degree} \\ \\ so \\ \\ sin \theta = sin45 = \frac{1}{ \sqrt{2} } \\ \\ cosec 45= \frac{1}{sin45} = \sqrt{2} \\ \\and....... \\ \\ cos45 = \frac{1}{ \sqrt{2} } \\ \\ sec45 = \frac{1}{cos45} = \sqrt{2}$

Answer 2

Question:

If $\tan\:=\:\sf\:1$, then find the value of $\sf\:\dfrac{\sin\:\theta\:+\:\cos\:\theta}{\sec\:\theta\:+\:\csc\:\theta}$

Answer:

$\boxed{\red{\sf\:\dfrac{\sin\:\theta\:+\:\cos\:\theta}{\sec\:\theta\:+\:\csc\:\theta}\:=\:\dfrac{1}{2}}}$

Step-by-step-explanation:

We have given that, $\tan\:\theta\:=\:\sf\:1$

From trigonometry table, we know that,

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

We have to find the value of

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

Now,

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

$\displaystyle\sf\:=\:\dfrac{2}{\sqrt{2}}\:\div\:2\:\sqrt{2}\\\\\\\sf\:=\:\dfrac{\cancel2}{\sqrt{2}}\:\times\:\dfrac{1}{\cancel{2}\:\sqrt{2}}\\\\\\\sf\:=\:\dfrac{1}{\sqrt{2}\:\times\:\sqrt{2}}\\\\\\\sf\:=\:\dfrac{1}{(\:\sqrt{2}\:)^2}\\\\\\\sf\:=\:\dfrac{1}{2}\\\\\\\therefore\boxed{\red{\sf\:\dfrac{\sin\:\theta\:+\:\cos\:\theta}{\sec\:\theta\:+\:\csc\:\theta}\:=\:\dfrac{1}{2}}}$

Additional Information:

$\begin{array}{|c|c|c|c|c|c|}\cline{1-6}\multicolumn{6}{|c|}{\sf\:Trigonometric\:Table}\\\cline{1-6}\multicolumn{1}{|c|}{\sf\:Trigonometric\:ratio} & \multicolumn{5}{|c|}{\sf\:Angle\:(\:\theta\:)}\\\cline{1-6}& \sf\:0^{\circ} & \sf\:30^{\circ} & \sf\:45^{\circ} & \sf\:60^{\circ} & \sf\:90^{\circ}\\\cline{1-6}\sin\:\theta & \sf\:0 & \sf\dfrac{1}{2} & \sf\dfrac{1}{\sqrt{2}} & \sf\dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}\cos\:\theta & \sf\:1 & \sf\dfrac{\sqrt{3}}{2} & \sf\dfrac{1}{\sqrt{2}} & \sf\dfrac{1}{2} & 0\\\cline{1-6}\tan\:\theta & \sf0 & \sf\dfrac{1}{\sqrt{3}} & \sf1 & \sf\sqrt{3} & \sf\:Not\:de fined\\\cline{1-6}\end{array}$ $\begin{array}{|c|c|c|c|c|c|}\cline{1-6}\multicolumn{6}{|c|}{\sf\:Trigonometric\:Table}\\\cline{1-6}\multicolumn{1}{|c|}{\sf\:Trigonometric\:ratio} & \multicolumn{5}{|c|}{\sf\:Angle\:(\:\theta\:)}\\\cline{1-6}& \sf\:0^{\circ} & \sf\:30^{\circ} & \sf\:45^{\circ} & \sf\:60^{\circ} & \sf\:90^{\circ}\\\cline{1-6}\csc\:\theta\:=\:\sf\:\dfrac{1}{\sin\:\theta} & \sf\:Not\:de fined & \sf\:2 & \sf\:\sqrt{2} & \sf\:\dfrac{2}{\sqrt{3}} & \sf\:1\\\cline{1-6}\sec\:\theta\:=\:\dfrac{1}{\cos\:\theta} & \sf\:1 & \sf \dfrac{2}{\sqrt{3}} & \sf \sqrt{2} & \sf 2 & \sf Not\ de fined\\\cline{1-6}\cot\:\theta\:=\:\dfrac{1}{\tan\:\theta} & \sf Not\ de fined & \sf \sqrt{3} & \sf 1 & \sf \dfrac{1} {\sqrt{3}} & \sf 0\\\cline{1-6}\end{array}$