Find the Value of cos²76° + cos²16° − cos76° cos16°
Question
Find the value of
\[ \cos^2 76^\circ+\cos^2 16^\circ-\cos76^\circ\cos16^\circ. \]Solution
Since
\[ 76^\circ+14^\circ=90^\circ, \]we have
\[ \cos76^\circ=\sin14^\circ. \]Let
\[ a=\cos76^\circ,\qquad b=\cos16^\circ. \]Using the identity
\[ a^2+b^2-ab=(a-b)^2+ab, \]and the product-to-sum formula,
\[ \cos76^\circ\cos16^\circ = \frac{\cos60^\circ+\cos92^\circ}{2}. \] \[ = \frac{\frac12-\sin2^\circ}{2}. \]Also,
\[ \cos^2 76^\circ+\cos^2 16^\circ = \frac{1+\cos152^\circ}{2} + \frac{1+\cos32^\circ}{2}. \] \[ = 1+\frac{\cos152^\circ+\cos32^\circ}{2}. \] \[ = 1+\frac{-\cos28^\circ+\cos32^\circ}{2}. \]Combining all terms and simplifying gives
\[ \cos^2 76^\circ+\cos^2 16^\circ-\cos76^\circ\cos16^\circ = \frac{3}{4}. \]