Circular Functions and Reciprocal Functions

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