MHB How to Prove this Trigonometric Identity?

AI Thread Summary
The discussion focuses on proving the trigonometric identity involving sine and cosine functions. By letting θ equal 5¾°, the problem simplifies to proving that the sum of three sine-to-cosine ratios equals half the difference of two tangent functions. The key step involves establishing the identity that links the sine and cosine ratios to the tangent functions. The proof is constructed by substituting θ, 3θ, and 9θ into the derived identity, ultimately confirming the original statement. The final step is to prove the foundational identity, which remains open for further exploration.
anemone
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Prove $$\frac{\sin\left(5\tfrac{3}{4}^{\circ} \right)}{\cos\left(17\tfrac{1}{4}^{\circ} \right)}+\frac{\sin\left(17\tfrac{1}{4}^{\circ} \right)}{\cos\left(51\tfrac{3}{4}^{\circ} \right)}+\frac{\sin\left(51\tfrac{3}{4}^{\circ} \right)}{\cos\left(155\tfrac{1}{4}^{\circ} \right)}=\frac{1}{2}\left(\tan\left(155\tfrac{1}{4}^{\circ} \right)-\tan\left(5\tfrac{3}{4}^{\circ} \right) \right)$$
 
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anemone said:
Prove $$\frac{\sin\left(5\tfrac{3}{4}^{\circ} \right)}{\cos\left(17\tfrac{1}{4}^{\circ} \right)}+\frac{\sin\left(17\tfrac{1}{4}^{\circ} \right)}{\cos\left(51\tfrac{3}{4}^{\circ} \right)}+\frac{\sin\left(51\tfrac{3}{4}^{\circ} \right)}{\cos\left(155\tfrac{1}{4}^{\circ} \right)}=\frac{1}{2}\left(\tan\left(155\tfrac{1}{4}^{\circ} \right)-\tan\left(5\tfrac{3}{4}^{\circ} \right) \right)$$
Let $\theta = 5{\frac34}^\circ$. Then the problem becomes
Prove $$\frac{\sin\theta}{\cos(3\theta)} + \frac{\sin(3\theta)}{\cos(9\theta)} + \frac{\sin(9\theta)}{\cos(27\theta)} = \tfrac12\bigl(\tan(27\theta) - \tan\theta\bigr)$$.
Suppose we knew that $$\frac{\sin x}{\cos(3x)} = \tfrac12\bigl(\tan(3x) - \tan x\bigr).\quad(*)$$

Putting $x$ equal to $\theta$, then $3\theta$, and then $9\theta$, it would follow that $$\begin{aligned}\frac{\sin\theta}{\cos(3\theta)} + \frac{\sin(3\theta)}{\cos(9\theta)} + \frac{\sin(9\theta)}{\cos(27\theta)} &= \tfrac12\bigl(\tan(3\theta) - \tan\theta\bigr)+ \tfrac12\bigl(\tan(9\theta) - \tan(3\theta)\bigr)+ \tfrac12\bigl(\tan(27\theta) - \tan(9\theta)\bigr) \\ &= \tfrac12\bigl(\tan(27\theta) - \tan\theta\bigr), \end{aligned}$$ as required. It just remains to prove the identity (*), which I'll leave for someone else.
 
Opalg said:
Let $\theta = 5{\frac34}^\circ$. Then the problem becomes

Suppose we knew that $$\frac{\sin x}{\cos(3x)} = \tfrac12\bigl(\tan(3x) - \tan x\bigr).\quad(*)$$

Putting $x$ equal to $\theta$, then $3\theta$, and then $9\theta$, it would follow that $$\begin{aligned}\frac{\sin\theta}{\cos(3\theta)} + \frac{\sin(3\theta)}{\cos(9\theta)} + \frac{\sin(9\theta)}{\cos(27\theta)} &= \tfrac12\bigl(\tan(3\theta) - \tan\theta\bigr)+ \tfrac12\bigl(\tan(9\theta) - \tan(3\theta)\bigr)+ \tfrac12\bigl(\tan(27\theta) - \tan(9\theta)\bigr) \\ &= \tfrac12\bigl(\tan(27\theta) - \tan\theta\bigr), \end{aligned}$$ as required. It just remains to prove the identity (*), which I'll leave for someone else.

we have tan 3x - tan x = sin 3x/ cos 3x - sin x/ cos x
= ( sin 3x cos x - cos 3x sin x)/ ( cos 3x sin x)
= sin 2x/( cos3x sin x)
= ( 2 sin x cos x)/(cos 3x sin x)
= 2 sin x / cos 3x
or sin x/ cos 3x = 1/2( tan 3x - tan x)
 
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