Question

$\sin ^{2} 75 + \sin ^{2} 15$
​

Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\large{\boxed{\sf{sin ^2\left(75^{\circ \:}\right)+sin ^2\left(15^{\circ \:}\right)}}}$

♣ ᴀɴꜱᴡᴇʀ :

$\large{\boxed{\sf{sin ^2\left(75^{\circ \:}\right)+sin ^2\left(15^{\circ \:}\right)=1}}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\mathrm{Use\:the\:following\:identity}:\quad \sin \left(x\right)=\cos \left(90^{\circ \:}-x\right)$

$\sin \left(75^{\circ \:}\right)=\cos \left(90^{\circ \:}-75^{\circ \:}\right)$

$=\cos ^2\left(90^{\circ \:}-75^{\circ \:}\right)+\sin ^2\left(15^{\circ \:}\right)$

$=\cos ^2\left(15^{\circ \:}\right)+\sin ^2\left(15^{\circ \:}\right)$

$\begin{array}{l}\cos \left(15^{\circ}\right)=\dfrac{\sqrt{2+\sqrt{3}}}{2} \\\\\\\sin \left(15^{\circ}\right)=\dfrac{\sqrt{2-\sqrt{3}}}{2}\end{array}$

$\sf{\cos ^2\left(15^{\circ \:}\right)+\sin ^2\left(15^{\circ \:}\right)=\left(\dfrac{\sqrt{2+\sqrt{3}}}{2}\right)^2+\left(\dfrac{\sqrt{2-\sqrt{3}}}{2}\right)^2}$

$\huge\boxed{\sf{=1}}$