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:

 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)


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


Rebecca L. GCSE Maths tutor, A Level Maths tutor, GCSE Physics tutor,...
View profile
£20 /hr

Rebecca L.

Degree: Physics (Bachelors) - Manchester University

Subjects offered: Maths, Physics+ 1 more


“I am studying physics at the University of Manchester. I have been involved in volunteering to organise events for elderly people within my local community. As well as this I have aided in delivering workshops to schools in my local a...”

MyTutor guarantee

PremiumRyan B. GCSE Chemistry tutor, A Level Chemistry tutor, GCSE Physics t...
View profile
£24 /hr

Ryan B.

Degree: Natural Sciences (Masters) - Durham University

Subjects offered: Maths, Physics+ 1 more


“I am currently a 1st year student at Durham University studying Natural Sciences. I have always had apassion for science and believe that developing your interest in the subject, during these sessions, will be the key to your success....”

Jonathan B. A Level Chemistry tutor, GCSE Chemistry tutor, A Level Ma...
View profile
£20 /hr

Jonathan B.

Degree: Chemistry (Masters) - Oxford, Lincoln College University

Subjects offered: Maths, Physics+ 1 more


“Hi, my name is Jonny and I have just finished my first year studying Chemistry at Oxford University, taking modules inChemistry, Physics and Maths. I have experience tutoring secondary school maths and science students and I have a pa...”

About the author

PremiumSam  G. A Level Maths tutor, A Level Further Mathematics  tutor, GCSE...
View profile
£24 /hr

Sam G.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Maths, Science+ 3 more

Further Mathematics

“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

How do I know which method of integration to use?

How do i differentiate the equation y = x^2 + 6x + 2 with respect to x.

How do you integrate the function cos^2(x)

If y = (1+3x)^2, what is dy/dx?

View A Level Maths tutors


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