Question

Simplify cosθ + sinθtanθ.

Answer 1

You should already know that:

$tan\theta = \frac{sin\theta}{cos\theta}$

Plugging that in we get:

$cos\theta + sin\theta (\frac{sin\theta}{cos\theta}) =\\\\cos\theta +\frac{sin^2\theta}{cos\theta}$

Multiply the cos theta by cos theta on the numerator and denominator so we can add them :)

$(cos\theta \times \frac{cos\theta}{cos\theta})+\frac{sin^2\theta}{cos\theta} = \\\\\frac{cos^2\theta}{cos\theta} + \frac{sin^2\theta}{cos\theta}=\\\\\frac{cos^2\theta + sin^2\theta}{cos\theta}$

You should also know the identity that cos^2 theta + sin^2 theta = 1

$\frac{cos^2\theta + sin^2\theta}{cos\theta} = \\\\\frac{1}{cos\theta}$

Therefore,

$cos\theta + sin\theta tan\theta = \boxed{\frac{1}{cos\theta}}$