Question

if $\sec \theta +\tan \theta =2+\sqrt{5}$,the the value of $\sin \theta +\cos \theta$ is:

• A. $\frac{3}{\sqrt{5}}$
• B. $\sqrt{5}$
• C. $\frac{2}{\sqrt{5}}$
• D. $-\frac{3}{\sqrt{5}}$

Answer 1

Answer: (A)

$\frac{3}{\sqrt{5}}$

$sec\Theta +tan\Theta =2+\sqrt{5}$______(1)

Also $se{c}^2\Theta -ta{n}^2\Theta =1$

$\therefore sec\Theta -tan\Theta =\frac{1}{2+\sqrt{5}}$

$\therefore sec\Theta -tan\Theta =-(2-\sqrt{5})$

$\therefore sec\Theta -tan\Theta =\sqrt{5}-2$______(2)

$\therefore$ From (1) & (2)

$2sec\Theta =2\sqrt{5}$

$\therefore sec\Theta =\sqrt{5}$

$\therefore cos\Theta =\frac{1}{\sqrt{5}}$

$\therefore sin\Theta =\frac{2}{\sqrt{5}}$

$\therefore sin\Theta +cos\Theta =\frac{3}{\sqrt{5}}$.