Even and Odd Trigonometry Functions
All functions, including trigonometry
functions, can be described as being even, odd, or neither.
A function is said to be odd if
and only if f(x) =  f(x) at all values of x and is symmetric with respect to
the origin.
Similarly, function is even if
and only if f(x) = f(x) at all values of x and is symmetric to the y axis.
It is helpful to know if a function is odd or
even when we are trying to simplify an expression when the variable inside the
trigonometric functions are negative.
Sin( Ө ) =  sin Ө

Cosec
( Ө ) =  cosec Ө

Cos ( Ө ) = cos Ө

Sec (Ө ) = sec Ө

Tan ( Ө ) =  tan Ө

Tan ( Ө ) =  tan Ө

Proof of Even Odd
Trigonometric Function:
From the definition of Cosine(Cos)
and Sine(Sin) in Unit Circle,
Put x = CosӨ
and y = SinӨ,
We can see that for both Ө and –Ө in the above
given figure. Value of “x” remains same, Hence Cos(Ө) = Cos(Ө).
But, it is clearly seen that the value of y for Ө and – Ө are different an
additive inverse , thus Sin(Ө) is not equals Sin(Ө), Here Sin(Ө)= Sin Ө
Similarly we can prove other trigonometric
functions,
Tan(Ө)

=

Sin(Ө)

=

Sin(Ө)

=

Tan(Ө)

Cos(Ө)

Cos(Ө)


Cot(Ө)

=

1

=

1

=

Cot(Ө)

Tan(Ө)

TanӨ


Cosec(Ө)

=

1

=

1

=

Cosec(Ө)

Sin(Ө)

Sin(Ө)


Sec(Ө)

=

1

=

1

=

Sec(Ө)

Cos(Ө)

Cos(Ө)

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