Trigonometry

Proof of the reciprocal identities

Proof of the tangent and cotangent identities

Proof of the Pythagorean identities

The proof of each of those follows from the definitions of the trigonometric functions, Topic 15.

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  =   y
x
.

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  =  1².  .  .  .  .  .  .  .  .  .  .  .  .  .(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 cos²θ and 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.

Trigonometric identities


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