Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 206471 by MATHEMATICSAM last updated on 15/Apr/24

$$\mathrm{If}a\sin \theta =b\cos \theta =\frac{2c\tan \theta }{1-{\tan }^2\theta }\mathrm{then}\mathrm{prove}$$$$\mathrm{that}{(a^2-b^2)}^2=4c^2(a^2+b^2).$$

Answered by lepuissantcedricjunior last updated on 15/Apr/24

$$\mathit{a}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }=\frac{2\mathit{c}\mathit{t}\mathit{a}\mathit{n}\mathit{\theta }}{1-{\mathit{t}\mathit{a}\mathit{n}}^2\mathit{\theta }}=\mathit{c}\mathit{t}\mathit{a}\mathit{n}2\mathit{\theta }=\frac{2\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }}{2{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{\theta }-1}=\mathit{b}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$$$$\iff \mathit{a}=\frac{2\mathit{c}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }}{2{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{\theta }-1};\mathit{b}=\frac{2\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }}{2{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{\theta }-1}$$$$\mathit{o}\mathit{n}\mathit{a}$$$${({\mathit{a}}^2-{\mathit{b}}^2)}^2={\left(\frac{4{\mathit{c}}^2\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }}{{(2{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{\theta }-1)}^2}\right)}^2=16{\mathit{c}}^4{\left(\frac{\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }}{{\mathit{c}\mathit{o}\mathit{s}}^2\left(2\mathit{\theta }\right)}\right)}^2$$$$\Rightarrow {({\mathit{a}}^2-{\mathit{b}}^2)}^2=16{\mathit{c}}^4\left(\frac{1}{{\mathit{c}\mathit{o}\mathit{s}}^2\left(2\mathit{\theta }\right)}\right)$$$$=16{\mathit{c}}^4\left[\frac{1}{4{\mathit{c}}^2}(\frac{4{\mathit{c}}^2{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{\theta }}{{\mathit{c}\mathit{o}\mathit{s}}^2\left(2\mathit{\theta }\right)}+\frac{4{\mathit{c}}^2{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{\theta }}{{\mathit{c}\mathit{o}\mathit{s}}^2\left(2\mathit{\theta }\right)})\right]$$$$=4{\mathit{c}}^2({\mathit{a}}^2+{\mathit{b}}^2)$$$$\Rightarrow {({\mathit{a}}^2-{\mathit{b}}^2)}^2=4{\mathit{c}}^2({\mathit{a}}^2+{\mathit{b}}^2)$$$$.........\mathit{l}\mathit{e}\mathit{p}\mathit{u}\mathit{i}\mathit{s}\mathit{s}\mathit{a}\mathit{n}\mathit{t}\mathit{c}\mathit{e}\mathit{d}\mathit{r}\mathit{i}\mathit{c}\mathit{j}\mathit{u}\mathit{n}\mathit{i}\mathit{o}\mathit{r}.....$$$$$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com