Take LHS:
\[{{\left( cos\text{ }\alpha \text{ }+\text{ }cos\text{ }\beta \right)}^{2}}~+\text{ }{{\left( sin\text{ }\alpha \text{ }+\text{ }sin\text{ }\beta \right)}^{2}}\]
By expanding we get,
\[{{\left( cos\text{ }\alpha \text{ }+\text{ }cos\text{ }\beta \right)}^{2}}~+\text{ }{{\left( sin\text{ }\alpha \text{ }+\text{ }sin\text{ }\beta \right)}^{2}}~=\]
\[=\text{ }co{{s}^{2}}~\alpha \text{ }+\text{ }co{{s}^{2}}~\beta \text{ }+\text{ }2\text{ }cos\text{ }\alpha \text{ }cos\text{ }\beta \text{ }+\text{ }si{{n}^{2}}~\alpha \text{ }+\text{ }si{{n}^{2}}~\beta \text{ }+\text{ }2\text{ }sin\text{ }\alpha \text{ }sin\text{ }\beta \]
\[=\text{ }2\text{ }+\text{ }2\text{ }cos\text{ }\alpha \text{ }cos\text{ }\beta \text{ }+\text{ }2\text{ }sin\text{ }\alpha \text{ }sin\text{ }\beta \]
\[=\text{ }2\text{ }\left( 1\text{ }+\text{ }cos\text{ }\alpha \text{ }cos\text{ }\beta \text{ }+\text{ }sin\text{ }\alpha \text{ }sin\text{ }\beta \right)\]
\[=\text{ }2\text{ }\left( 1\text{ }+\text{ }cos\text{ }\left( \alpha \text{ }\text{ }\beta \right) \right)\text{ }\left[ since,\text{ }cos\text{ }\left( A\text{ }\text{ }B \right)\text{ }=\text{ }cos\text{ }A\text{ }cos\text{ }B\text{ }+\text{ }sin\text{ }A\text{ }sin\text{ }B \right]\]
\[=\text{ }2\text{ }(1\text{ }+\text{ }2\text{ }co{{s}^{2}}~\left( \alpha \text{ }\text{ }\beta \right)/2\text{ }\text{ }1)\text{ }[since,\text{ }cos2x\text{ }=\text{ }2co{{s}^{2}}~x\text{ }\text{ }1]\]
\[=\text{ }2\text{ }(2\text{ }co{{s}^{2}}~\left( \alpha \text{ }\text{ }\beta \right)/2)\]
\[=\text{ }4\text{ }co{{s}^{2}}~\left( \alpha \text{ }\text{ }\beta \right)/2\]
= RHS
Hence Proved.