the equation is not an identity, however (after playing with the calculator) ...
for
for equal to integral multiples of and odd multiples of
for
for equal to integral multiples of and odd multiples of
for
for equal to integral multiples of and odd multiples of
hmmm ...
conjecture that for equal to integral multiples of and odd multiples of
... ???
It needs the use of three trig identities.
Briefly, take a difference of two squares on the LHS, simplify the two brackets using (1) and (2) and then use (3) (twice) to simplify the resulting expression. Move the resulting expression to the RHS, remove as a common factor and then use (2) again on the expression within the brackets.
That leads to
from which or or