Trigonometry XI


   
 
Other Circular Functions (Reciprocal functions of cosq, sinq and tanq)
 
 
 
 
 
 
 
Theorem 16
 
Statement:
 
(i) 1 + tan2 x = sec2x
 
(ii) 1 + cot2 x = cosec2x
 
Proof:
 
Consider the following identity
 
 
Divide by cos2x throughout, then
 
 
 
 
Divide by sin2x throughout, then
 
 
 
Theorem 17
 
Statement:
 
 
 
 
 
 
 
Proof
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Theorem 18
 
 
Proof:
 
Let n > 0. When n = 0, tan (0p + x) = tan x which is true. The theorem is proved by mathematical induction when n=1, tan (p+x)=tan p=RHS.
 
Let the theorem be true for n = m > 0 then tan(mp + x) = tan p
 
 
 
Since the theorem is true for all n = 1, n = m, n = m+1 when m > 0
 
theorem is true for n > 0.
 
Next when n < 0. Let n = -m where m > 0
 
 
 
 
 
Theorem 19
 
 
Proof:
 
 
 
 
 
[Dividing both Numerator and denominator by cosx cosy]
 
 
Theorem 20
 
 
Proof:
 
 
 
 
 
 
Theorem 21
 
 
 
 
Proof:
 
 
 
 
 
 
 
 
Theorem 22
 
 
 
 
Proof:
 
 
 
[Dividing both Numerator and denominator by ]
 
 
 
 
 
 
 
Theorem 23
 
Prove that
 
 
 
Proof:
 
 
 
 
 
 
 
 
 
 
 
 
Theorem 24
 
 
 
 
Proof:
 
(i) sin[(x+y) + z]
 
 
 
 
 
(ii) cos[(x+y) + z]
 
 
 
 
 
 
Dividing (1) by (2), we get
 
 
  
     
   
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