Question

If cosec x-cot x=1/3 where x is not equal to zero then the value of cos²x-sin²x is ______

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}, \\ \mathrm{cosec}\mathrm{x}-\cot \mathrm{x}=\frac{1}{3} \\ \mathrm{Since}\mathrm{cosec}\mathrm{x}=\frac{1}{\sin \mathrm{x}}\mathrm{and}\cot \mathrm{x}=\frac{\cos \mathrm{x}}{\sin \mathrm{x}},\mathrm{so}\mathrm{it}\mathrm{implies}\mathrm{that}, \\ \frac{1}{\sin \mathrm{x}}-\frac{\cos \mathrm{x}}{\sin \mathrm{x}}=\frac{1}{3} \\ \frac{1-\mathrm{cosx}}{\sin \mathrm{x}}=\frac{1}{3} \\ \mathrm{Since}2{\sin }^2\mathrm{\theta }=1-\cos 2\mathrm{\theta }\Rightarrow 2{\sin }^2\frac{\mathrm{\theta }}{2}=1-\mathrm{cos\theta } \\ \mathrm{and}\sin 2\mathrm{\theta }=2\mathrm{cos\theta sin\theta }\Rightarrow \mathrm{sin\theta }=2\cos \frac{\mathrm{\theta }}{2}\sin \frac{\mathrm{\theta }}{2} \\ \mathrm{This}\mathrm{implies}\mathrm{that} \\ \frac{2{\sin }^2{\displaystyle \frac{\mathrm{x}}{2}}}{2\sin {\displaystyle \frac{\mathrm{x}}{2}}\cos \frac{\mathrm{x}}{2}}=\frac{1}{3} \\ \frac{\sin {\displaystyle \frac{\mathrm{x}}{2}}}{{\displaystyle \cos \frac{\mathrm{x}}{2}}}=\frac{1}{3} \\ \tan \frac{\mathrm{x}}{2}=\frac{1}{3} \\ \mathrm{The}\mathrm{value}\mathrm{of}\cos \mathrm{x}\mathrm{and}\sin \mathrm{x}\mathrm{can}\mathrm{be}\mathrm{found}\mathrm{as}, \\ \cos \mathrm{x}=\frac{1-{\tan }^2\frac{\mathrm{x}}{2}}{1+{\tan }^2\frac{\mathrm{x}}{2}}=\frac{1-{\displaystyle \frac{1}{9}}}{1+{\displaystyle \frac{1}{9}}}=\frac{{\displaystyle \frac{8}{9}}}{{\displaystyle \frac{10}{9}}}=\frac{8}{10}=\frac{4}{5} \\ \sin \mathrm{x}=\frac{2\tan \frac{\mathrm{x}}{2}}{1+{\tan }^2\frac{\mathrm{x}}{2}}=\frac{{\displaystyle \frac{2}{3}}}{1+{\displaystyle \frac{1}{9}}}=\frac{{\displaystyle \frac{2}{3}}}{{\displaystyle \frac{10}{9}}}=\frac{2}{3}\times \frac{9}{10}=\frac{3}{5} \\ \mathrm{So}\mathrm{the}\mathrm{required}\mathrm{value}\mathrm{is}, \\ {\cos }^2\mathrm{x}-{\sin }^2\mathrm{x}={\left(\frac{4}{5}\right)}^2-{\left(\frac{3}{5}\right)}^3=\frac{16}{25}-\frac{9}{25}=\frac{7}{25} \end{gathered}$$