You should already know that:
[tex]tan\theta = \frac{sin\theta}{cos\theta}[/tex]
Plugging that in we get:
[tex]cos\theta + sin\theta (\frac{sin\theta}{cos\theta}) =\\\\cos\theta +\frac{sin^2\theta}{cos\theta}[/tex]
Multiply the cos theta by cos theta on the numerator and denominator so we can add them :)
[tex](cos\theta \times \frac{cos\theta}{cos\theta})+\frac{sin^2\theta}{cos\theta} = \\\\\frac{cos^2\theta}{cos\theta} + \frac{sin^2\theta}{cos\theta}=\\\\\frac{cos^2\theta + sin^2\theta}{cos\theta}[/tex]
You should also know the identity that cos^2 theta + sin^2 theta = 1
[tex]\frac{cos^2\theta + sin^2\theta}{cos\theta} = \\\\\frac{1}{cos\theta}[/tex]
Therefore,
[tex]cos\theta + sin\theta tan\theta = \boxed{\frac{1}{cos\theta}}[/tex]
