Question

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Answer 1

Answer:

$\sf{\dfrac{11}{6}}$

Values Used:

$\sf{\sin30^\circ=\dfrac{1}{2}}\\\\\sf{\sin60^\circ=\dfrac{\sqrt{3}}{2}}\\\\\sf{\cos45^\circ=\dfrac{1}{\sqrt{2}}}\\\\\sf{\cos60^\circ=\dfrac{1}{2}}\\\\\sf{\tan60^\circ=\sqrt{3}}$

Step-by-step explanation:

$\sf{4(\sin^430^\circ+\cos^460^\circ)-\dfrac{2}{3}(\sin^260^\circ-\cos^245^\circ)+\dfrac{1}{2}\tan^260^\circ}$

$\sf{=4\left(\left(\dfrac{1}{2}\right)^4+\left(\dfrac{1}{2}\right)^4\right)-\dfrac{2}{3}\left(\left(\dfrac{\sqrt{3}}{2}\right)^2-\left(\dfrac{1}{\sqrt{2}}\right)^2\right)+\dfrac{1}{2}\left(\sqrt{3}\right)^2}$

$\sf{4\left(\dfrac{1}{16}+\dfrac{1}{16}\right)-\dfrac{2}{3}\left(\dfrac{3}{4}-\dfrac{1}{2}\right)+\dfrac{3}{2}}$

$\sf{4\times\dfrac{2}{16}-\dfrac{2}{3}\times\dfrac{1}{4}+\dfrac{3}{2}}$

$\sf{\dfrac{1}{2}-\dfrac{1}{6}+\dfrac{3}{2}}$

$\sf{2-\dfrac{1}{6}}$

$\sf{\dfrac{12-1}{6}}$

$\boxed{\sf{\dfrac{11}{6}}}$