How do you simplify #f(theta)=cot4theta-sin2theta-2cos8theta# to trigonometric functions of a unit #theta#?

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Answer 1 · 2018-08-02T12:57:39

See answer in the explanation.

Use expansions forming real and imaginary parts in the

expansions

from # ( cos theta + i sin theta )^n= cos ntheta + i sin ntheta, n = 2, 4 and 8#.

For brevity, #c = cos theta and s = sin theta#.

#f = f_1 + f_2 + f_3#,

#f_1 = cot 4theta = cos (4theta)/sin (4theta)#

#= ( c^4-6 c s ( c^2 - s^2 ) + s^4 )/(4 c s ( c^2 - s^2 ))#

#f_2 = - sin 2theta = - 2 c s#

#f_3 = - 2 cos 8theta#

#= - 2 (c^8 - 28 c s (c^6 - s^6 ) + 70 c^2 s^2 ( c^2 - s^2) +s^8 )#

#f = f_1 + f_2 + f_3#.

Of course, simplification is possible, using

#(c^2 + s^2 )^m = 1, m = 1, 2, 3, 4.#