Question

Find the value of sin2 π/10+sin2 4π/10+sin2 6π/10+sin2 9π/10

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO EVALUATE

$\displaystyle \sf{ { \sin}^{2} \: \frac{\pi}{10} \: + \: { \sin}^{2} \: \frac{4\pi}{10} + \:{ \sin}^{2} \: \frac{6\pi}{10} \: + \: { \sin}^{2} \: \frac{9\pi}{10} \: }$

CALCULATION

$\displaystyle \sf{ { \sin}^{2} \: \frac{\pi}{10} \: + \: { \sin}^{2} \: \frac{4\pi}{10} + \:{ \sin}^{2} \: \frac{6\pi}{10} \: + \: { \sin}^{2} \: \frac{9\pi}{10} \: }$

$= \displaystyle \sf{ { \sin}^{2} \: \frac{\pi}{10} \: + \: { \sin}^{2} \: \frac{4\pi}{10} + \:{ \sin}^{2} \bigg(\pi \: - \frac{4\pi}{10} \bigg) + \:{ \sin}^{2} \bigg(\pi \: - \frac{\pi}{10} \bigg) \: }$

$= \displaystyle \sf{ { \sin}^{2} \: \frac{\pi}{10} \: + \: { \sin}^{2} \: \frac{4\pi}{10} + { \sin}^{2} \: \frac{\pi}{10} \: + \: { \sin}^{2} \: \frac{4\pi}{10} \: }$

$= \displaystyle \sf{2 \bigg( { \sin}^{2} \: \frac{\pi}{10} \: + \: { \sin}^{2} \: \frac{4\pi}{10} \bigg) \: }$

$= \displaystyle \sf{2 \bigg[ { \sin}^{2} \: \frac{\pi}{10} \: + \: { \sin}^{2} \bigg( \frac{\pi}{2} \: - \frac{\pi}{10} \bigg) \bigg] \: }$

$= \displaystyle \sf{2 \bigg[ { \sin}^{2} \: \frac{\pi}{10} \: + \: { \cos}^{2} \frac{\pi}{10} \bigg] \: }$

$= \displaystyle \sf{2 \times 1\: }$

$= \displaystyle \sf{2 }$

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