Question

$\mathrm{If}\tan \theta =1,\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\sin \theta +\cos \theta }{sec\theta +\cos ec\theta },\mathrm{where}\mathrm{\theta }\mathrm{is}\mathrm{an}\mathrm{acute}\mathrm{angle}.$

Answer 1

$$\begin{gathered} \tan \theta =1\Rightarrow cot\theta =1 \\ \mathrm{We}\mathrm{know}\mathrm{that}1+{\tan }^2\theta ={\sec }^2\theta \\ \Rightarrow 1+1^2=se{c}^2\theta \\ \Rightarrow se{c}^2\theta =2 \\ \Rightarrow sec\theta =\sqrt{2} \\ \therefore \cos \theta =\frac{1}{sec\theta }=\frac{1}{\sqrt{2}} \\ \mathrm{Also},1+co{t}^2\theta =\cos e{c}^2\theta \\ \Rightarrow 1+1^2=\cos e{c}^2\theta \\ \Rightarrow \cos ec\theta =\sqrt{2} \\ \mathrm{Now},\sin \theta =\frac{1}{\cos ec\theta }=\frac{1}{\sqrt{2}} \\ \therefore \frac{\sin \theta +\cos \theta }{sec\theta +\cos ec\theta }=\frac{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}{\sqrt{2}+\sqrt{2}}=\frac{1}{2} \end{gathered}$$