Question

if sin theta - cos theta =1/2 then find the value of 1/ sin theta + cos theta

Answer 1

I have join an attachment

Answer 2

Thus, The answer is $\frac{2\sqrt{7} }{7}$

Step-by-step explanation:

Given,

$sin\theta -cos \theta=\frac{1}{2}$ then $\frac{1}{sin\theta+cos\theta} =?$

∴ $sin\theta -cos \theta=\frac{1}{2}$

⇒$(sin\theta -cos \theta)^{2} =(\frac{1}{2})^{2}$  (Squaring both sides)

⇒$sin^{2}\theta+cos^{2}\theta-2.sin\theta.cos\theta=\frac{1}{4}$

⇒$1-\frac{1}{4}=2.sin\theta.cos\theta$   (∵$sin^{2}\theta+cos^{2}\theta=1$)

⇒$\frac{3}{4}=2.sin\theta.cos\theta$

⇒$sin\theta.cos\theta=\frac{3}{8}$

∴ $(sin\theta+cos\theta)^{2} =sin^{2}\theta+cos^{2}\theta+2.sin\theta.cos\theta$

                         $=1+2\times \frac{3}{8}$

                         $=\frac{4+3}{4}$

                         $=\frac{7}{4}$

⇒$sin\theta +cos \theta=\sqrt{\frac{7}{4} }$

⇒$sin\theta +cos \theta=\frac{\sqrt{7} }{2}$

∴$\frac{1}{sin\theta+cos\theta} =\frac{1}{\frac{\sqrt{7} }{2} }$

                $=\frac{2}{\sqrt{7} }$

                $=\frac{2\sqrt{7} }{\sqrt{7}\times \sqrt{7} }$

                $=\frac{2\sqrt{7} }{7}$

∴ The answer is $\frac{2\sqrt{7} }{7}$