PROVING IDENTITIES TRIGONOMETRY 4. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Proving identities 1/ 7 Adrian Jannetta

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1 PROVING IDENTITIES TRIGONOMETRY 4 INU05/55 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Proving identities / 7 Adrian Jannetta

2 Proving an identity Proving an identity is a process which starts with the (left-hand side) or the RHS (right-hand side) of a given relationship and, through a series of logical steps, shows that the other side of the relationship can be obtained. Proving an identity Prove the identity(x+) 3 x 3 + 3x 2 + 3x+ In this example, we can start with the and make the following arguments. They may look different but the right side and left sides of an identity are the same. (x+) 3 (x+)(x+) 2 (x+)(x 2 + 2x+) x(x 2 + 2x+)+x 2 + 2x+ x 3 + 2x 2 + x+x 2 + 2x+ x 3 + 3x 2 + 3x+ RHS Proving identities 2/ 7 Adrian Jannetta

3 Proving a trig identity Prove the identity secθ tanθ sinθ+ Which side do we begin changing? We ll choose the RHS because there more things we can change. RHStanθ sinθ+ We know that tanθ sinθ so we substitute this for tanθ : RHS Put the terms over a common denominator: sinθ sinθ+ sin 2 θ + RHS sin2 θ+ cos 2 θ But know that sin 2 θ+ cos 2 θ so the top simplifies: RHS RHS secθ Proving identities 3/ 7 Adrian Jannetta

4 Proving a trig identity Prove that( )(+secα)sinαtanα. Let s start with the and make changes to it. ( )(+secα) +secα secα +secα +secα secα Remember we are aiming to have an expression with sinα and tanα. cos 2 α sin 2 α sinα sinα sinαtanα RHS Proving identities 4/ 7 Adrian Jannetta

5 Useful strategies... There are no rules which for proving trig identities that will work in every situation. But we can talk about strategies or guidelines that might help. Start with more complicated side of the identity Start changing things! e.g. tanθ can be expressed as sinθ /. e.g. tan 2 θ can be expressed as sin2 θ/cos 2 θ and so on. Reciprocal trig ratios can be changed; e.g. secθ= / Look for the fundamental identity sin 2 θ+ cos 2 θ=. Or variations of it; sin 2 θ can be replaced with cos 2 θ. Look for Pythagorean identities; e.g. sec 2 θ =tan 2 θ. Be aware of other identities that might simplify things; e.g. sin2θ can be expressed as 2sin θ. If fractions are present put them over a common denominator Cancel and simplify if possible. This list is not exhaustive and we simply can t list all the ways you might prove an identity! Proving identities 5/ 7 Adrian Jannetta

6 Proving a trig identity Prove that tan x+cotx sec x cosecx. Starting with the Multiply top and bottom by cosx: tanx+cotx secx cosec x cos x + cos x cos x cos x + cos x sin x cosx sin x cosx cos x sin2 x+cos 2 x RHS Proving identities 6/ 7 Adrian Jannetta

7 Final comments... In principle it s possible to start with either side of the identity and make changes until the other side has been reached. = RHS or RHS= In practice it is often more obvious how to begin the proof with one side of the identity than it is to begin with the other side. This is something that gets easier to recognise through experience (doing more examples!) Alternatively, we might begin with one side of the identity and change it a little. Then, perhaps we go to the other side of the identity and try to change it enough to meet halfway. = = RHS This approach is like building a bridge from opposite sides of a river with the aim of meeting in the middle. If this works then the path from to RHS (or vice versa) will be obvious at the end of the proof. Finally which results do we assume to be true to help us prove identities? For now, we can take the Pythagorean Identities to be proved since they were proved using a right-angled triangle in earlier in the course. Proving identities 7/ 7 Adrian Jannetta