The question is to find the value of $ \csc \theta + \cot \theta $ if $\sec \theta + \tan \theta = 5$ . Here is what I did : $\sec \theta + \tan \theta = 5$
$\sec \theta = 5 - \tan \theta $
Squaring both sides , $$\sec^2 \theta = 25 + \tan^2 \theta -10\tan \theta$$ Substituting $1+\tan^2 \theta$ for $\sec^2 \theta$ , $$1+\tan^2 \theta = 25 + \tan^2 \theta -10\tan \theta$$ Thus , $$\tan \theta=24/10$$ So , $\cot \theta = 10/24 $ and $\csc \theta=26/24$
Thus $ \csc \theta + \cot \theta =3/2$ . But I checked the answer sheet and the answer is not 3/2 but $(3+\sqrt5 )/2$ . Where have I went wrong ? Please help.