(i)
(ii)
Solution:
(i)
![\[1\text{ }=\text{ }RHS\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-182a87bd389cab2b4ec180aaed688b55_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)
![\[\left\{ 1+cotx\left-( -cosecx \right) \right\}\left\{ 1+cotx+\left( -cosecx \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b7b50b6a767c39833d6d25363b67e3e2_l3.png "Rendered by QuickLaTeX.com")
![\[\left\{ 1\text{ }+\text{ }cot\text{ }x\text{ }+\text{ }cosec\text{ }x \right\}\text{ }\left\{ 1\text{ }+\text{}cot\text{ }x\text{}-\text{}cosec\text{}x\right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b8752edfec6e03c58fa53326ecdda272_l3.png "Rendered by QuickLaTeX.com")
![\[\left\{ \left( 1\text{ }+\text{ }cot\text{ }x \right)\text{ }+\text{ }\left( cosec\text{ }x \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4ad83859c4dfe392ef79b6f2dbe36b50_l3.png "Rendered by QuickLaTeX.com")
![\[\left\{ \left( 1\text{ }+\text{ }cot\text{ }x \right)\text{ }-\text{ }\left( cosec\text{ }x \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dfd19a8007995eca60eba9edd57def61_l3.png "Rendered by QuickLaTeX.com")
By using the formula,
![\[\left( a\text{ }+\text{ }b \right)\text{ }\left( a\text{ }-\text{ }b \right)\text{ }=\text{ }{{a}^{2}}-\text{ }{{b}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-55b7c271946377a6d110c05dbe005a52_l3.png "Rendered by QuickLaTeX.com")
![\[{{\left( 1\text{ }+\text{ }cot\text{ }x \right)}^{2}}-\text{ }{{\left( cosec\text{ }x \right)}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4f46d0adee1e8e058b3ff26c6e98a43b_l3.png "Rendered by QuickLaTeX.com")
![\[1\text{ }+\text{ }co{{t}^{2~}}x\text{ }+\text{ }2\text{ }cot\text{ }x\text{ }-\text{ }cose{{c}^{2~}}x\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-586829f65a7e6649a9f38fb51c963836_l3.png "Rendered by QuickLaTeX.com")
We know that
![\[1\text{ }+\text{ }co{{t}^{2~}}x\text{ }=\text{ }cose{{c}^{2~}}x\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-181b2a51c416b2cb4640c276b3723a61_l3.png "Rendered by QuickLaTeX.com")
![\[cose{{c}^{2~}}x\text{ }+\text{ }2\text{ }cot\text{ }x\text{ }-\text{ }cose{{c}^{2~}}x\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-02fe24fec239079cacf759569a6b1343_l3.png "Rendered by QuickLaTeX.com")
![\[2\text{ }cot\text{ }x\text{ }=\text{ }RHS\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-56095c8e0d9bd59e6736863ffdcba187_l3.png "Rendered by QuickLaTeX.com")
Hence proved.
(i)
(ii)
Solution:
(i)
Hence proved.
(ii)
By using the formula,
We know that
Hence proved.