The six functions used in trigonometry are sine, cosine, tangent, cotangent, secant and cosecant. But there are thousands of relationships that exist among these functions. Proving a trigonometric relationship requires imaginative problem solving skills and algebraic factoring. For those that enjoy mathematics, as well as solving puzzles, you'll feel a sense of accomplishment verifying identities.
What are the fundamental trigonometric identities? The reciprocal
identities are the facts that the reciprocal of sine, cosine and tangent
are the three trigonometric functions cosecant, secant and cotangent,
respectively. Therefore, it holds true that the reciprocal of cosecant,
secant and cotangent are sine, cosine and tangent.
Since sine is opposite divided by hypotenuse and cosine is adjacent
divided by hypotenuse and tangent is opposite divided by adjacent, it
holds true that tangent is sine divided by cosine. Therefore, cotangent
is cosine divided by sine.
Recall the Pythagorean Theorem which states that the sum of the square
of the legs of a right triangle equals the hypotenuse squared (a2 + b2 = c2). From this relationship we know that sin2x + cos2x = 1, 1 + tan2x = sec2x and 1 + cot2x = csc2x.
The pythagorean identities can be proven, but it's more important to
know the identities, as they are used frequently in verifying other
Let's verify some identities by changing to sines and cosines. That is
often a method used to verify identities. Always work from the side of
the equation that is most complicated, proving it equals the side of the
equation that is less complex.
For example: Verify that (cscx)(tanx) = secx.
Change every function in terms of sine and cosine. Therefore we have
(1/sinx)(sinx/cosx) = secx
Notice the sinx in the denominator and the sinx in the numerator
cancel each other, so we are left with 1/cosx and we know from the
reciprocal identities that it equals secx, so we have verified the
Let's try a more complex example.
Verify that (cosx)(cotx) + sinx = cscx
Again, work with the left side of the equation since it's the more
complex side and change everything in terms of sine and cosine.
Therefore, we get
cosx(cosx/sinx) + six = cscx
cos2x/sinx + sinx = cscx
Getting a common denominator of sin2x, the equation becomes
(cos2x + sin2x)/sinx = cscx
Notice that cos2x + sin2x = 1, therefore the equation becomes 1/sinx which we know from the reciprocal identities is cscx. The identity is verified!
Let's try one more example. This is the most difficult of the three.
Verify that sinx/(1 + cosx) = (1- cosx)/sinx
The method used in the previous example does not apply here, since
everything is already in terms of sine and cosine. So now we have to get
creative and find out how we can use a pythagorean identity to solve
this. The pythagorean identities have squared terms in them, in
particular in this case sin2x + cos2x = 1. Notice we can maneuver the equation to get 1 - sin2x = cos2x and 1 - cos2x = sin2x.
If we work on the left side of the equation and multiply the numerator
and denominator by 1 + cosx, we will get a desired result.
sinx/(1 + cosx) * (1 - cosx)/(1 - cosx) = sinx(1 - cosx)/(1 - cos2x)
Notice the 1 - cos2x in the denominator, which is perfect
as it matches up with the manipulated version of the pythagorean
identity above. The value of 1 - cos2x = sin2x. The equation now becomes sinx(1 - cosx)/sin2x.
The sinx in the numerator cancels with a sinx in the denominator,
leaving us with (1 - cosx)/sinx. The identity is now verified!
There are many examples of verifying identities in any trigonometry or
pre-calculus book. Learn the identities and strategies for manipulating
equations. Practice and you'll become a master and solving such