Proof Of The Reciprocal Relations

 

By definition:

sin θ

  =  

y
r

csc θ

  =  

 

r
y

Therefore, sin θ is the reciprocal of csc θ:

sin θ

  =  

   1   
csc θ

where 1-over any quantity is the symbol for its reciprocal; Lesson 5 of Algebra. Similarly for the remaining functions.

Proof of the tangent and cotangent identities

To prove:

tan θ  =  

sin θ
cos θ

and

 

cot θ  =  

cos θ
sin θ

.

Proof By definition,

tan θ  =  

.

Therefore, on dividing both numerator and denominator by r,

tan θ

  =  

y/r
x/r

  =  

sin θ
cos θ

.

 

 

cot θ

  =  

   1   
tan θ

  =  

cos θ
sin θ

.

Those are the two identities.

Proof of the Pythagorean identities

To prove:

 

a)

sin2θ + cos2θ

  =  

1

 

b)

1 + tan2θ

  =  

sec2θ

 

c)

1 + cot2θ

 

csc 2θ

Proof 1: According to the Pythagorean theorem,

x2 + y2 = r2.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .(1)

Therefore, on dividing both sides by r2,

x2
r2

  +  

y2
r2

  =  

r2
r2

  =  1.

That is, according to the definitions,

cos2θ  +  sin2θ  =  1.  .  .  .  .  .  .  .  .  .  .  .  .  .(2)

Apart from the order of the terms, this is the first Pythagorean identity, a).

To derive b), divide line (1) by x2; to derive c), divide by y2.

Or, we can derive both b) and c) from a) by dividing it first by cos2θand then by sin2θ.  On dividing line 2) by cos2θ, we have

That is,

1 + tan2θ  =  sec2θ.

And if we divide a) by sin2θ, we have

That is,

1 + cot2θ  =  csc2θ.

The three Pythagorean identities are thus equivalent to one another.

  Proof 2.

 

sin2θ  + cos2θ 

y2
r2

+

x2
r2

 

 

=

y2 + x2
   r2

 

 

=

r2
r2

According to the Pythagorean theorem,

 

=

1.