MYTUTOR SUBJECT ANSWERS

509 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.

Sam  G. A Level Maths tutor, A Level Further Mathematics  tutor, GCSE...

2 years ago

Answered by Sam , an A Level Maths tutor with MyTutor


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

296 SUBJECT SPECIALISTS

£20 /hr

Jack G.

Degree: Politics, Philosophy, and Economics (Bachelors) - Warwick University

Subjects offered:Maths, Religious Studies+ 2 more

Maths
Religious Studies
Economics
.TSA. Oxford.

“Hello, I'm Jack and am studying politics, philosophy, and economics at Warwick.  Having recently been through the subjects I'm looking to tutor I feel like I know first hand what difficulties you'll be facing. I always found I wanted ...”

MyTutor guarantee

£22 /hr

Raimonds B.

Degree: Architecture (Bachelors) - Bath University

Subjects offered:Maths, Physics

Maths
Physics

“I am a first year Architecture student, as well as a well experienced tutor. I have tutored GCSE and A Level Mathematics and Physics for over two years, and have 5 star reviews on First Tutors. I have an exam based approach to revision...”

£20 /hr

Rachel B.

Degree: Mathematics with French (Bachelors) - Manchester University

Subjects offered:Maths, French

Maths
French

“I'm a friendly, enthusiastic and passionate student tutor, dedicated to spreading my love of Maths and French.”

MyTutor guarantee

About the author

PremiumSam  G. A Level Maths tutor, A Level Further Mathematics  tutor, GCSE...

Sam G.

Currently unavailable: for regular students

Degree: Mathematics (Bachelors) - Durham University

Subjects offered:Maths, Science+ 3 more

Maths
Science
Physics
Further Mathematics
Chemistry

“Hi, I'm Sam, a mathematics student at Durham. Although I will mainly focus on A level and GCSE if people have other questions or more challenging questions (perhaps STEP or MAT) I will be more than happy to help!”

You may also like...

Posts by Sam

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

The point A lies on the curve y=5(x^2)+9x , The tangent to the curve at A is parralel to the line 2y-x=3. Find an equation to this tangent at A.

How do we differentiate y=a^x when 'a' is an non zero real number

A curve has the equation (x+y)^2 = xy^2. Find the gradient of the curve at the point where x=1

What methods are there for integration?

View A Level Maths tutors

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

mtw:mercury1:status:ok