What is the value of $$\cos \theta$$?
I. $$\sec \theta + \tan \theta$$ = 5
II. $$1 + \sin \theta = \frac{25}{13}$$

I.

we know that $$\sec^2\theta\ -\tan^2\theta\ =1.$$

or,$$\left(\sec\theta\ -\tan\theta\ \right)\left(\sec\theta\ +\tan\theta\ \right)=1.$$

here,$$\sec\theta\ +\tan\theta\ =5.$$

So,$$\sec\theta\ -\tan\theta\ =1\div5=0.2.$$

So, $$2\sec\theta\ =5+0.2=5.2.$$

or,$$\sec\theta\ =2.6.$$

or,$$\cos\theta\ =\frac{\ 10}{26}=\frac{\ 5}{13}.$$

II.

we know that $$\sin^2\theta\ +\cos^2\theta\ =1.$$

or,$$\cos\theta\ =\pm\ \sqrt{1-\sin^2\theta\ \ }.$$

here,$$\sin\theta\ =\ \frac{\ 12}{13}.$$

So,$$\cos\theta\ =\pm\ \sqrt{1-\ \frac{\ 144}{169}\ }=\pm\ \sqrt{\ \frac{\ 25}{169}\ }=\ \pm\ \frac{\ 5}{13}.$$

So, Option A is correct choice.