Trigonometric identity of finite terms Prove that: 1/(cos x+cos 3x) + 1/(cos x+ cos (5x))+...+1/(cos x+ cos ((2n+1)x))=1/2 csc x [ tan((n+1)x)-\tan(x)] I tried to prove this using the regular formulas. But failed. Please help me.

cjortiz141t

cjortiz141t

Answered question

2022-09-05

Trigonometric identity of finite terms
Prove that:
1 cos x + cos 3 x + 1 cos x + cos 5 x + + 1 cos x + cos ( 2 n + 1 ) x = 1 2 csc x [ tan ( n + 1 ) x tan x ]
I tried to prove this using the regular formulas. But failed. Please help me.

Answer & Explanation

Azul Lang

Azul Lang

Beginner2022-09-06Added 20 answers

k = 1 n 1 cos x + cos ( 2 k + 1 ) x = k = 1 n 1 2 cos k x cos ( k + 1 ) x =
= 1 2 sin x k = 1 n ( tan ( k + 1 ) x tan k x ) = 1 2 sin x ( tan ( n + 1 ) x tan x )
and we are done!
I used the following reasoning.
tan α tan β = sin α cos α sin β cos β = sin α cos β cos α sin β cos α cos β = sin ( α β ) cos α cos β .
genestesya

genestesya

Beginner2022-09-07Added 1 answers

In the case n=0 there is no summand on the left-hand side and the right-hand side is 0, so the base case of the induction holds.
Suppose it holds for n−1; then the left-hand side with n can be written, by the induction hypothesis,
1 2 sin x ( tan n x tan x ) + 1 cos x + cos ( 2 n + 1 ) x
and you want to prove this equals
1 2 sin x ( tan ( n + 1 ) x tan x )
which is equivalent to
tan n x 2 sin x + 1 cos x + cos ( 2 n + 1 ) x = tan ( n + 1 ) x 2 sin x
By the sum-to-product formulas, this becomes
sin n x 2 sin x cos n x + 1 2 cos n x cos ( n + 1 ) x = sin ( n + 1 ) x 2 sin x cos ( n + 1 ) x
Reduce to the same denominator and conclude the equality holds.
The relation
sin n x cos ( n + 1 ) x + sin x = sin ( n + 1 ) x cos n x
is true, because it is equivalent to
sin x = sin ( n + 1 ) x cos n x cos ( n + 1 ) x sin n x

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