# Simplify tan$\theta$ cos$\theta$

How do I simplify tan$\theta$ cos$\theta$ ?

Why is this so hard to do? What pieces of information should I know before doing these?

Can someone just tell me were am I going wrong? I have 5 days to master this before my SAT Practice.

I would brush up on 'basics' but i don't even know how this related to anything ive learnt in trig before...I literally google every identity, am i supposed to remember or somehow derive them??

• Do you remember the definition of tan? – Mathmo123 Jul 22 '14 at 10:27
• Do you know how tan relates to sin and cos? (4 days, 23h, 58min) – Lord_Gestalter Jul 22 '14 at 10:28
• sosmath.com/trig/Trig5/trig5/trig5.html – Scientifica Jul 22 '14 at 10:28
• To use the 5 days brushing up the basics will be worth more than collecting and memorising possible answers – Lord_Gestalter Jul 22 '14 at 10:34
• I saw your earlier posts. You are going wrong, by not concentrating on the basic definition of sine, cos and tan. If I was you, I would take a pen and paper and start by drawing triangle and learn the ratios of sin,cos and tan. It would not be called maths, if it was not tricky! :) – MonK Jul 22 '14 at 10:35

## 4 Answers

Hint: What is the definition of the $\tan$ function?

• Tan = Opp / Adj.. – sasha Jul 22 '14 at 10:38
• @sasha That is a rough definition only true for angles from $0$ to $\frac\pi2$. For other corners, the $\tan$ is defined via the $\cos$ and $\sin$ function. If you are practising for SATs, I advise you to look that definition up. – 5xum Jul 22 '14 at 10:40
• @sasha $\tan\theta = opp/adj = \displaystyle\frac{opp}{adj}\cdot\frac{\frac{1}{hyp}}{\frac{1}{hyp}} = \frac{\frac{opp}{hyp}}{\frac{opp}{hyp}}=\frac{\sin\theta}{\cos\theta}$ – John Joy Jul 22 '14 at 13:30

Notice first that the $\tan$ function is defined on $D:=\Bbb R\setminus\{\frac\pi2+k\pi,\; k\in\Bbb Z\}$ and that $\tan\theta=\frac{\sin\theta}{\cos\theta}$ so

$$\tan\theta\cos\theta=\sin\theta,\quad\forall\theta\in D$$ and be careful the trap is to give a wrong domain.

• I am used to write - rather than $\$ for excluding. Anyway +1 – mrs Jul 22 '14 at 12:21

Elementary school proof for $\theta \in ]0, \pi/2[$:

Consider a right triangle with angles $\pi/2, \theta, \pi/2-\theta$ and edges lengths $a,b,c>0$ as in the figure below

Then $$\tan(\theta)= \frac{a}{c}, \quad \cos(\theta) = \frac{c}{b}, \quad \sin(\theta) = \frac{a}{b},$$ thus $$\tan(\theta) \cos(\theta) = \frac{a}{c}\cdot \frac{c}{b} = \frac{a}{b} = \sin(\theta).$$

There are two ways to do this. Here is one way:

Saying that

$$\tan(x)=o/a$$ and $$\cos(x)=a/h$$, we get $$oa/ha = o/h=\sin x$$ so the answer is $$\sin x$$!