Derivation Of Pythagorean Identities

Derivation of Pythagorean Identities:

1. The illustration shown above helps support the derivation of the two Pythagorean identities in which we come up with. To help explain this process of deriving I am going to begin by explaining how the pythagorean theorem is derived from the unit circle as shown above. So as we know from our previous knowledge of the unit circle we are allowed to draw a right triangle. This allows for us to then label the sides of the triangle with different variables, which leads us to write the pythagorean theorem a^2+b^2=c^2. Being that we know that this triangle is within the Unit Circle we can use previous knowledge of how the hypotenuse will equal 1 which in this case is represented by the letter c. In this example, the side of a will be equal to cosine theta and side b will be equal to sine of theta. After figuring out which variable stands for what trig function we can see how the pythagorean identity is derived, because the values are substituted into the original equation (a^2+b^2=c^2) to produce the identity which is cosine theta^2+sin of theta^2=1. It is easy to follow how this equation was derived from the original pythagorean theorem in the illustration above. Then, to derive the other identities from the Pythagorean identity all it requires you to do is divide. For the next pythagorean identity what you have to do is divide the original equation by cosine^2 theta. As shown in the illustration above, when doing this process in number 2 the cos^2theta from the original equation will cancel out to just leaving a 1 and the sin^2 theta divided by a cos ^2 theta could be substituted in as tan^2 theta. We can also see how the 1 divided by cos^2 theta turns into sec^2 theta, which is because it is a reciprocal identity. When gathering up this information and putting the equation back together it is easy to understand how we come up with the identity to being 1+tan^2 theta=sec^2 theta. The last pythagorean identity follows the same process except now we are dividing by sin^2 theta from the original Pythagorean identity. When dividing the entire equation by sin^2 theta we are able to recognize other identities, which allow us to substitute such as cos^2 theta/sin^2 theta that equals cot^2 theta. After getting this we can easily see in example 3 how the sin^2 theta cancels out to leave a +1, and how csc^2 theta gets substituted in for 1/sin^2 theta which is a reciprocal identity. Once we have simplified this equation, we then are left with our third pythagorean identity which is cot^2 theta+1=csc^2 theta.