Question Number 11844 by minakshidahaval0202@gmail.co. last updated on 02/Apr/17 $$\mathrm{cot}\alpha+\mathrm{cosec}\alpha={k}\:{then}\:{find}\:\mathrm{cosec}\alpha−{cot}\alpha\:{and}\:{also}\:{find}\:{cot}\alpha \\ $$ Answered by sma3l2996 last updated on 02/Apr/17 $${we}\:{have} \\ $$$$\left({i}\right):{cot}\alpha+{cosec}\alpha=\frac{{cos}\alpha}{{sin}\alpha}+\frac{\mathrm{1}}{{sin}\alpha}={k} \\ $$$$=\frac{{cos}\alpha+\mathrm{1}}{{sin}\alpha}=\frac{\left({cos}\alpha+\mathrm{1}\right)\left({cos}\alpha−\mathrm{1}\right)}{{sin}\alpha\left({cos}\alpha−\mathrm{1}\right)}=\frac{−{sin}^{\mathrm{2}} \alpha}{{sin}\alpha\left({cos}\alpha−\mathrm{1}\right)}…
Question Number 11843 by tawa last updated on 02/Apr/17 $$\mathrm{Differentiate},\:\:\mathrm{ln}\left(\mathrm{cosx}\right)\:\:\:\mathrm{from}\:\mathrm{the}\:\mathrm{first}\:\mathrm{principle}. \\ $$ Answered by ajfour last updated on 02/Apr/17 $$\frac{{d}}{{dx}}\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right)=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\mathrm{cos}\:\left({x}+{h}\right)−\mathrm{ln}\:\mathrm{cos}\:{x}}{{h}} \\ $$$$\:=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left[\:\frac{\mathrm{cos}\:\left({x}+{h}\right)}{\mathrm{cos}\:{x}}\:\right]}{{h}} \\…
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Question Number 142914 by loveineq last updated on 07/Jun/21 $$\mathrm{Let}\:{a}\geqslant{b}\geqslant{c}\geqslant{d}>\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}+{d}\:=\:\mathrm{4}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}}{\mathrm{3}}\:\leqslant\:\frac{\mathrm{1}}{\:\sqrt{{d}}} \\ $$$$\mathrm{Prove}\:\mathrm{if}\:\forall{n}\in\mathbb{N}^{+} ,\:\mathrm{then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt[{{n}}]{{a}}+\sqrt[{{n}}]{{b}}+\sqrt[{{n}}]{{c}}}{\mathrm{3}}\:\leqslant\:\frac{\mathrm{1}}{\:\sqrt[{{n}}]{{d}}} \\ $$$$ \\ $$ Terms of…
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Question Number 11838 by Peter last updated on 02/Apr/17 $$\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}^{\mathrm{2}} −\mathrm{1}}\:+\:\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{1}}{\mathrm{3}^{\mathrm{2}} −\mathrm{1}}\:+\:\frac{\mathrm{4}^{\mathrm{2}} +\mathrm{1}}{\mathrm{4}^{\mathrm{2}} −\mathrm{1}}\:+\:….\:+\:\frac{\mathrm{20}^{\mathrm{2}} +\mathrm{1}}{\mathrm{20}^{\mathrm{2}} −\mathrm{1}}\:=\:….? \\ $$ Answered by ajfour last updated…
Question Number 142910 by BHOOPENDRA last updated on 07/Jun/21 Answered by qaz last updated on 07/Jun/21 $$\mathrm{y}\left(\mathrm{t}\right)=\mathrm{t}+\mathrm{e}^{−\mathrm{2t}} +\int_{\mathrm{0}} ^{\mathrm{t}} \mathrm{y}\left(\tau\right)\mathrm{e}^{\mathrm{2}\left(\mathrm{t}−\tau\right)} \mathrm{d}\tau \\ $$$$\mathscr{L}=\frac{\mathrm{1}}{\mathrm{s}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{s}+\mathrm{2}}+\frac{\mathscr{L}}{\mathrm{s}−\mathrm{2}}…………..\mathscr{L}=\mathscr{L}\left(\mathrm{y}\left(\mathrm{t}\right)\right)\left(\mathrm{s}\right) \\…
Question Number 142904 by liberty last updated on 07/Jun/21 $${If}\:\mathrm{2}+\mathrm{log}\:_{\mathrm{2}} \left({x}\right)=\mathrm{3}+\mathrm{log}\:_{\mathrm{3}} \left({y}\right)=\mathrm{log}\:_{\mathrm{6}} \left({x}−\mathrm{4}{y}\right) \\ $$$${then}\:{the}\:{value}\:{of}\:\frac{\mathrm{1}}{\mathrm{2}{y}}−\frac{\mathrm{2}}{{x}}=? \\ $$ Answered by bramlexs22 last updated on 07/Jun/21 Terms…