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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

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