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This page contains sample problems -- identities high school students have been asked to prove.

Proving an Identity

Suppose you are asked to prove

(sin(x)-cos(x)+1) / (sin(x)+cos(x)-1) = (sin(x)+1) / cos(x)

The quickest way to convince yourself it's true is to cross-multiply:

(sin - cos + 1) cos = (sin + cos - 1) (sin + 1) 
sin cos - cos2 + cos = sin2 + sin cos - sin + sin + cos - 1
sin cos - cos2 + cos = sin cos + sin2 - 1 + cos
sin cos - cos2 + cos = sin cos - cos2 + cos

But does that prove the identity?  No.  What we've done here is assumed the identity, and then we used it to derive another more obvious identity.  So it's not a proof at all.  Although the method of cross-multiplying is very convincing, and usually disproves a false "identity", it does not prove the identity here, because we assumed the thing to be proved.

So many high school math teachers will simply not allow you to cross multiply as a method of solving this kind of problem.  Instead they insist that you start from the left (or right) side, and make algebraic and trigonometric manipulations until you arrive at the expression on the right (or left) side.  That's not always very easy, because it's not at all obvious what manipulations are needed.

Rescuing the cross-multiplying method

There is a way to rescue the "proof" above, and turn it into an actual proof.  It works by taking every step in reverse order:

sin cos - cos2 + cos = sin cos - cos2 + cos     (true because left = right)
sin cos - cos2 + cos = sin cos + sin2 - 1 + cos     (true because -cos2 = sin2 - 1)
sin cos - cos2 + cos = sin2 + sin cos - sin + sin + cos - 1     (true because sin-sin=0, and I just moved things around)
(sin - cos + 1) cos = (sin + cos - 1) (sin + 1)     (true because I factored out cos on the left and sin on the right) 
(sin - cos + 1) / (sin + cos - 1) = (sin + 1) / cos     (true by dividing both sides by cos and by (sin + cos - 1) )

This is a valid proof.  However, your teacher may still disapprove.  Maybe you're pulling a fast one!  Here's a plan "B" for you to try.

What to do if your teacher still disapproves

The identity given here, like so many you will encounter, is of the form

A / B = C / D

We can obey the teacher's rule by starting with A/B, and multiplying both the numerator and denominator by CD, giving us

A / B = (ACD) / (BCD)

Now, by cross-multiplying, we have shown that AD = BC, so we can substitute BC in place of AD in the numerator, giving us

     = (BC2) / (BCD)

And then we can cancel common factors, giving

     = C / D

"Linear combination of fractions" method: If A/B = C/D, then (rA+sC)/(rB+sD) = C/D

That is, if we form a linear combination of the two numerators, and divide that by the same linear combination of the two denominators, you will get yet another fraction that is equal to A/B and also to C/D.

Consider again, the problem

(sin(x)-cos(x)+1) / (sin(x)+cos(x)-1) = (sin(x)+1) / cos(x)

If we let A/B be cos(x) / (1-sin(x)) and C/D be (1+sin(x)) / cos(x)

You can check that A/B = C/D by cross-multiplying, or by whatever "approved" method your teacher provides!  (See "geometric progression identities" below for a whole bunch of similar cases)

So now, our problem boils down to

(C-A) / (D-B), which is equal by this "linear combination" method to C/D, proving our identity.

Proof: If A/B = C/D, then (rA+sC)/(rB+sD) = C/D

First, since A/B = C/D, we multiply both sides by B/C to derive

A/C = B/D

Now, we let k=A/C, so Ck=A and Dk=B, which gives us

(rA+sC)/(rB+sD) = ((rk+s)C) / ((rk+s)D) = C/D

Geometric progression identities

We briefly saw one of these geometric progression identities, above, in the section "Linear combination of fractions".  It was:

 (1+sin(x)) / cos(x) = cos(x) / (1-sin(x))

I call this a "geometric progression identity" because the quantities 1+sin(x),  cos(x),  and 1-sin(x) are in geometric progression.  You can check that by multiplying the first by the last, and you'll get the square of the middle.  (1+sin(x))(1-sin(x)) = 1-sin2(x) = cos2(x).

Others in this category include

(1-cos(x)) / sin(x) = sin(x) / (1+cos(x)), and this in fact is equal to tan(x/2)

Proof: tan(x/2) = 2sin2(x/2) / (2 sin(x/2) cos(x/2)) = (1-cos2(x/2)+sin2(x/2))/sin(x) = (1-cos(x))/sin(x)

(sec(x)-1) / tan(x) = tan(x) / (sec(x)+1))

(sec(x)-tan(x)) = 1 / (sec(x)+tan(x))

(csc(x)-1) / cot(x) = cot(x) / (csc(x)+1))

(csc(x)-cot(x)) = 1/ (csc(x)+cot(x))

Note, too, that these identities can be easily disguised by moving cot(x) from the denominator making it tan(x) in the numerator, etc.

tan and cot identities

Tan and cot can be expanded to sin/cos and cos/sin, respectively, which will help you prove some identities.  For example,

tan(x)+cot(x) = 1/(sin(x) cos(x)) = 2/sin(2x)

I'll leave off the x's to make it easier to follow:

sin/cos + cos/sin = sin²/(sin cos) + cos²/(sin cos) = 1/(sin cos) = 1/(1/2 sin(2x) = 2/sin(2x)

example 2: tan(x)-cot(x) = -2 cot(2x)

sin/cos - cos/sin = sin²/(sin cos) - cos²/(sin cos) = (1-2 cos²)/(sin cos) = -cos(2x)/(1/2 sin(2x)) = -2 cot(2x)

Internet references

Summary of trig identities 

Related Pages in this website


The webmaster and author of this Math Help site is Graeme McRae.