Trigonometry -Proving trig identities expl

TRIGONOMETRY – PROVING TRIG
IDENTITIES EXPLAINED

P Denise Smith

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TRIGONOMETRY – PROVING TRIG IDENTITIES EXPLAINED
Proving an identity is very different in concept from solving an equation. Though you'll use many of the
same techniques, they are not the same, and the differences are what can cause you problems.
For instance, sin(x) = 1/csc(x) is an identity. To "prove" an identity, you have to use logical steps to show
that one side of the equation can be transformed into the other side of the equation. To prove an
identity, your instructor may have told you that you cannot work on both sides of the equation at the
same time. This is correct.
Since you'll be working with two sides of an equation, it might be helpful to introduce some notation, if
you haven't seen it before. The "left-hand side" of an equation is denoted by LHS, and the "right-hand
side" is denoted as RHS.

 Prove the identity
It's usually a safe bet to start working on the side that appears to be more complicated. In this case, that
would be the LHS. Another safe bet is to convert things to sines and cosines, and see where that leads.
So my first step will be to convert the cotangent and cosecant into their alternative expressions:

Now I'll flip-n-multiply:

Now I can see that the sines cancel, leaving me with:

Then my proof of the identity is all of these steps, put together:

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TRIGONOMETRY – PROVING TRIG IDENTITIES EXPLAINED
That final string of equations is what they're wanting for your answer.

 Prove the identity
I'm not sure which side is more complicated, so I'll just start on the left. My first step is to convert
everything to sines and cosines: Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

When I get fractions, it's almost always a good idea to get a common denominator, so I'll do that next:

so I'll do that next:

Now that I have a common denominator, I can combine these fractions into one:

Now I notice a Pythagorean identity in the numerator, allowing me to simplify:

Looking back at the RHS of the original identity, I notice that this denominator could be helpful. I'll
split the product into two fractions:

And now I can finish up by converting these fractions to their reciprocal forms:

(I wrote them in the reverse order, to match the RHS.) The complete answer is all of the steps
together, starting with the LHS and ending up with the RHS:

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TRIGONOMETRY – PROVING TRIG IDENTITIES EXPLAINED

If it’s easier for you you can use

SOH CAH TOA to solve for tan(x)….

Sin Cos Tan Learn the order of the top and
Csc Sec Cot bottom row and you will be
able to know all of the identites

1). tan(x) = opp/Adj
2). COT is the inverse of tan so cot(x) = adj/opp
3). Add together adj/opp + opp/adj (fraction rules say they have to have the same denominator)
4). So multiply your fraction on the left by (adj)/(adj) x adj/opp = adj2/(adj)(opp)
5). So multiply your fraction on the right by (opp)/(opp) x oppj/adj = opp2/(adj)(opp)
6) adj2/(adj)(opp) + opp2/(adj)(opp) (numerator is cos2+ sin2which is Pythagorean identity=1)
7) 1/opp x 1/adj = 1/sin x 1/cos
8). Inverse identity rule says 1/sin = cosecant and 1/cos = secant
9 ANSWER : csc(x) sec(x)

HAVE TO KNOW IDENTITY RULES
sin2+ cos2= 1 (pythagorean identities)
cot = cos/sin (quotient identity)
tan = sin/cos (inverse of cot)
1/sec = cos therefore 1/cos = sec (Inverse Identities)
1/sin = cosecant therefore 1/csc = sin (Inverse Identities)

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TRIGONOMETRY – PROVING TRIG IDENTITIES EXPLAINED

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