# Trigonometric Identity

#### anemone

##### MHB POTW Director
Staff member
Prove $$\displaystyle \frac{\sin\left(5\tfrac{3}{4}^{\circ} \right)}{\cos\left(17\tfrac{1}{4}^{\circ} \right)}+\frac{\sin\left(17\tfrac{1}{4}^{\circ} \right)}{\cos\left(51\tfrac{3}{4}^{\circ} \right)}+\frac{\sin\left(51\tfrac{3}{4}^{\circ} \right)}{\cos\left(155\tfrac{1}{4}^{\circ} \right)}=\frac{1}{2}\left(\tan\left(155\tfrac{1}{4}^{\circ} \right)-\tan\left(5\tfrac{3}{4}^{\circ} \right) \right)$$

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#### Opalg

##### MHB Oldtimer
Staff member
Prove $$\displaystyle \frac{\sin\left(5\tfrac{3}{4}^{\circ} \right)}{\cos\left(17\tfrac{1}{4}^{\circ} \right)}+\frac{\sin\left(17\tfrac{1}{4}^{\circ} \right)}{\cos\left(51\tfrac{3}{4}^{\circ} \right)}+\frac{\sin\left(51\tfrac{3}{4}^{\circ} \right)}{\cos\left(155\tfrac{1}{4}^{\circ} \right)}=\frac{1}{2}\left(\tan\left(155\tfrac{1}{4}^{\circ} \right)-\tan\left(5\tfrac{3}{4}^{\circ} \right) \right)$$
Let $\theta = 5{\frac34}^\circ$. Then the problem becomes
Prove $$\displaystyle \frac{\sin\theta}{\cos(3\theta)} + \frac{\sin(3\theta)}{\cos(9\theta)} + \frac{\sin(9\theta)}{\cos(27\theta)} = \tfrac12\bigl(\tan(27\theta) - \tan\theta\bigr)$$.
Suppose we knew that $$\displaystyle \frac{\sin x}{\cos(3x)} = \tfrac12\bigl(\tan(3x) - \tan x\bigr).\quad(*)$$

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.

Let $\theta = 5{\frac34}^\circ$. Then the problem becomes
Suppose we knew that $$\displaystyle \frac{\sin x}{\cos(3x)} = \tfrac12\bigl(\tan(3x) - \tan x\bigr).\quad(*)$$
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.