Proving Identities

Geometry Level pending

Which of the following is equivalent to sin ⁡ θ + tan ⁡ θ 1 + sec ⁡ θ \dfrac{\sin \theta + \tan \theta}{1 + \sec \theta} ?

sec ⁡ θ \sec \theta tan ⁡ θ \tan \theta cos ⁡ θ \cos \theta sin ⁡ θ \sin \theta

Marvin Kalngan by Marvin Kalngan , 38, Philippines

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

sin ⁡ θ + tan ⁡ θ 1 + sec ⁡ θ = sin ⁡ θ ( 1 + 1 cos ⁡ θ ) 1 + 1 cos ⁡ θ = sin ⁡ θ \frac {\sin \theta + \tan \theta}{1+\sec \theta} = \frac {\sin \theta \left(\cancel{1+\frac 1{\cos \theta}}\right)}{\cancel{1+\frac 1{\cos \theta}}} = \boxed{\sin \theta}

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Marvin Kalngan
May 1, 2020

We know that tan ⁡ θ = sin ⁡ θ cos ⁡ θ \tan \theta = \dfrac{\sin \theta}{\cos \theta} and sec ⁡ θ = 1 cos ⁡ θ \sec \theta = \dfrac{1}{\cos \theta} , so

sin ⁡ θ + tan ⁡ θ 1 + sec ⁡ θ \dfrac{\sin \theta + \tan \theta}{1 + \sec \theta}

= sin ⁡ θ + sin ⁡ θ cos ⁡ θ 1 + 1 cos ⁡ θ =\dfrac{\sin \theta + \dfrac{\sin \theta}{\cos \theta}}{1 + \dfrac{1}{\cos \theta}}

The LCD of the numerator is cos ⁡ θ \cos \theta . The LCD of the denominator is cos ⁡ θ \cos \theta . So

= sin ⁡ θ cos ⁡ θ + sin ⁡ θ cos ⁡ θ cos ⁡ θ + 1 cos ⁡ θ =\dfrac{\dfrac{\sin \theta \cos \theta + \sin \theta}{\cos \theta}}{\dfrac{\cos \theta + 1}{\cos \theta}}

sin ⁡ θ cos ⁡ θ + sin ⁡ θ cos ⁡ θ ÷ cos ⁡ θ + 1 cos ⁡ θ \dfrac{\sin \theta \cos \theta + \sin \theta}{\cos \theta} \div \dfrac{\cos \theta + 1}{\cos \theta}

= sin ⁡ θ cos ⁡ θ + sin ⁡ θ cos ⁡ θ × cos ⁡ θ cos ⁡ θ + 1 =\dfrac{\sin \theta \cos \theta + \sin \theta}{\cos \theta}\times \dfrac{\cos \theta}{\cos \theta + 1}

= sin ⁡ θ ( cos ⁡ θ + 1 ) cos ⁡ θ + 1 =\dfrac{\sin \theta (\cos \theta + 1)}{\cos \theta + 1}

= sin ⁡ θ =\color{#69047E}\boxed{\sin \theta} answer \boxed{\color{#3D99F6}\text{answer}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...