If cos 9$\alpha$ = sin$\alpha$ and 9$\alpha$

Question

If cos 9$\alpha$ = sin$\alpha$ and 9$\alpha$ < $90^\circ$, then the value of tan5$\alpha$ is ______.

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If cos 9$\alpha$ = sin$\alpha$ and 9$\alpha$ < $90^\circ$, then the value of tan5$\alpha$ is 1. Explanation: According to the question, cos 9$\alpha$ = sin $\alpha$ and 9$\alpha$ < $90^\circ$ i.e. 9$\alpha$ is an acute angle We know that, sin($90^\circ$ – $\theta$) = cos $\theta$ So, cos 9$\alpha$ = sin($90^\circ$ – $\alpha$) Since, cos 9$\alpha$ = sin($90^\circ$ – 9$\alpha$) and sin($90^\circ$ – $\alpha$) = sin $\alpha$ Thus, sin($90^\circ$ – 9$\alpha$) = sin $\alpha$ $90^\circ$ – 9$\alpha$ = $\alpha$ 10$\alpha$ = $90^\circ$ $\alpha$ = $9^\circ$ Substituting $\alpha$ = $9^\circ$ in tan 5$\alpha$, we get, tan 5$\alpha$ = tan(5 $\times$ $9^\circ$) = tan $45^\circ$ = 1 $\therefore$ tan 5$\alpha$ = 1

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