(ii)
Solution:
(i)
![\[co{{s}^{2}}A+co{{s}^{2}}B - 2cosAcosBcos\left(A+B \right)=si{{n}^{2}}\left(A+B\right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d9f89e6d0edf77a9253fab4102d7baa2_l3.png "Rendered by QuickLaTeX.com")
Let us consider LHS:
![\[co{{s}^{2}}A\text{ }+\text{ }co{{s}^{2}}B\text{}-\text{}2\text{}cos\text{}A\text{}cos\text{}B\text{}cos\text{}\left(A\text{}+B\right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-706679134e6b8fc1fc6397457064f583_l3.png "Rendered by QuickLaTeX.com")
![\[co{{s}^{2}}A\text{}+\text{}1\text{}-\text{}si{{n}^{2}}B\text{}-\text{}2\text{}cos\text{}A\text{}cos\text{}B\text{}cos\text{}\left(A\text{}+B \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e1d86baeffd537b75707c5786b52a52e_l3.png "Rendered by QuickLaTeX.com")
![\[1\text{}+co{{s}^{2}}A\text{}\text{}si{{n}^{2}}B\text{}-\text{}2\text}cos\text{}A\text{}cos\text{}B\text{}cos\text{}\left(A\text{}+B\right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b9da70ec67af0aa57b9bdede1ab13e72_l3.png "Rendered by QuickLaTeX.com")
Since,
![\[co{{s}^{2}}A\text{ }-\text{ }si{{n}^{2}}B\text{}=-\text{}cos\text{}\left(A\text{}+B\right)\text{}cos\text{ }\left( A\text{ }B \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-03499f8acf4d87919f39a3912d51727a_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\[1+~cos\left( A+B \right)cos\left( A-B \right)-2cosAcosBcos\left( A+B \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8992d7ae773908d88a3d710a3077ea2b_l3.png "Rendered by QuickLaTeX.com")
![\[1\text{ }+\text{ }cos\text{ }\left( A\text{ }+B \right)\text{ }\left[ cos\text{ }\left( A\text{ }-B \right)\text{ }-\text{ }2\text{ }cos\text{ }A\text{ }cos\text{ }B \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2a7cb13ffda69b7a24c5d07eca1d16bf_l3.png "Rendered by QuickLaTeX.com")
Since,
![\[cos\text{ }\left( A\text{ }-\text{ }B \right)\text{ }=\text{ }cos\text{ }A\text{ }cos\text{ }B\text{ }+\text{ }sin\text{ }A\text{ }sin\text{ }B.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2f64c89616dd02dba6966d40440ca0cc_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\[1+cos\left( A+B \right)\text{ }\left[ cosA\text{ }cosB+sinA\text{ }sinB\text{ }2\text{ }cosAcosB \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-85675d3de61caaf4e718b6d43c5cdb5b_l3.png "Rendered by QuickLaTeX.com")
![\[1\text{ }+\text{ }cos\text{ }\left( A\text{ }+B \right)\text{ }\left[ -cos\text{ }A\text{ }cos\text{ }B\text{ }+\text{ }sin\text{ }A\text{ }sin\text{ }B \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b684810c1a8d727c1c36f92ce2b7e0f0_l3.png "Rendered by QuickLaTeX.com")
![\[1\text{}+co{{s}^{2}}A\text{}-\text{}si{{n}^{2}}B\text{}-\text{}2\text{}cos\text{}A\text{}cos\text{}B\text{}cos\text{}\left(A\text{}+B\right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7e1c4bb00fb188844bf6ae86246aaf08_l3.png "Rendered by QuickLaTeX.com")
Since,
![\[cos\text{ }\left( A\text{ }+B \right)\text{ }=-\text{}cos\text{}A\text{}cos\text{}B\text{}-\text{}sin\text{}A\text{}sin\text{}B.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-547378bc6b4dbe18ce2ed6f372cf8109_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\[1\text{ }-\text{ }co{{s}^{2}}\left( A\text{ }+\text{ }B \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7c3dd2b89bb54acf07f4a06973c08cbe_l3.png "Rendered by QuickLaTeX.com")
![\[si{{n}^{2}}\left( A\text{ }+\text{ }B \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bf3b9815fe86fef3e367bf24fb6ddb65_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }RHS\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-357164290b65a799c74bee1fe89255df_l3.png "Rendered by QuickLaTeX.com")
![\[\therefore LHS\text{ }=\text{ }RHS\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ba03c21d058c733e389db8b909b2d82f_l3.png "Rendered by QuickLaTeX.com")
Hence proved.
(ii)
Hence proved.
(ii)
Solution:
(i)
Let us consider LHS:
Since,
Therefore,
Since,
Therefore,
Since,
Therefore,
Hence proved.
(ii)
Hence proved.