MYTUTOR SUBJECT ANSWERS

441 views

What is the best way to prove trig identities?

In my experience the sure fire way to prove trig identities is by doing the following:

1) Assess the question e.g. is it obvious you're going to need a double angle or addition formula, can you see where cos^2(x) + sin^2(x) = 1 would be needed etc.

2) Write down all the formulae you might need (it's also worth noting that two identities are easily proved by dividing cos^2(x) + sin^2(x) = 1 by cos^2(x) and another by sin^2(x) namely 1+tan^2(x) = sec^2(x) and cot^2(x) + 1 = csc^2(x).

3) Work from the more complicated side and reduce it to the simpler side. To prove that A = B is the same as proving B = A so it doesn't matter which way you start.

4) Some general tips if it's especially difficult. Try maybe factoring and seeing if a trig identity appears, try multiplying by 1 or adding 0 in "clever" ways. e.g. Maybe multiply by (sinx+1/sinx+1) then you haven't changed anything but it might be in a more useful form. Maybe also write simpler expressions as something else in case that's useful, e.g. instead of tanx write sinx/cosx. Finally if you have a fraction it might be worth multiplying both sides by the denominator and see if it's in a nicer form.

A little example I made up:

Show
 2sin^2(x) + 2cos^2(x) = (cos 2x)/(cos^2(x)) + sec^2(x) 

1) I can see I'll likely need the double angle formula for cos 2x but bear in mind there are 3 of those. I also notice on the left hand side there's no division and sinx and cosx are most familiar so I'm going to say the right hand side is harder so I'll start work from  there.

2) I see I've got cos(2x) so I might need cos^2(x) -  sin^2(x), I can also see I've got sec^2(x) so I might need tan^2(x) +1 = sec^2(x)

Let's begin.

RHS(right hand side) 

= cos(2x)/(cos^2(x)) + sec^2(x) (what we're given)

= (cos^2(x) - sin^2(x))/(cos^2(x))    + sec^2(x)     (expanding cos(2x) )

 = 1 - sin^2(x)/cos^2(x)     + sec^2(x)         ( carrying out division)

= 1 - tan^2(x) + sec^2(x)             ( realising sinx/cosx = tan(x))

= 1 + 1                                 (using sec^2(x) - tan^2(x) = 1)

=2 

But notice we can "cleverly" multiply by 1 (sin^2(x) + cos^2(x)) to get the desired result.

It's definitely worth noting that I went the long way round to try and exersize more techniques but the far better way to do this is to expand cos(2x) as 2cos^2(x) -1 because then you're immediately left with 2 after doing the division and cancelling the sec^2(x) which dramatically speeds up the process.

Expired account  . A Level Maths tutor, GCSE Maths tutor, A Level Fur...

1 year ago

Answered by Expired account, who tutored A Level Maths with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

253 SUBJECT SPECIALISTS

£20 /hr

Solomon L.

Degree: Mathematics (Bachelors) - Leeds University

Subjects offered: Maths, Further Mathematics

Maths
Further Mathematics

“Currently studying at Leeds university, ready to help you improve your grades in Maths and Chemistry GCSE! ”

£26 /hr

Josh R.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Maths, Further Mathematics

Maths
Further Mathematics

“I study Maths at Warwick, I hope to be able to teach your child to love Maths and how to achieve top exam grades. ”

£20 /hr

James B.

Degree: Mathematics and Economics (Bachelors) - Nottingham University

Subjects offered: Maths, Economics

Maths
Economics

“About me : I am currently studying Maths and Economics at Nottingham University, about to go into my second year. Over the summer I have been in America teaching and counselling kids at a summer camp and I hope to bring what I have le...”

About the author

£22 /hr

Expired account .

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Maths, Science+ 3 more

Maths
Science
Physics
Further Mathematics
Chemistry

“Hi, my name's Sam and I'm a mathematics (and physics) student at Durham. I know for some people the most important thing is just to get through their exams; I can andwill help you with this. I can't stress enough, however, how much e...”

You may also like...

Posts by Expired account

How do you take the derivative of a^x ?

How does integration work?

Some videos I've made

What is the best way to prove trig identities?

Other A Level Maths questions

How do you 'rationalise the denominator'?

differentiate y = (4-x)^2

Simplify (5-root3)/(5+root3)

Differentiate x^3(sinx) with respect to x

View A Level Maths tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok