Question

The exact value of cosec10 - 4sin70

Answer 1

$$\begin{gathered} \mathrm{cosec}10-4\sin 70 \\ =\frac{1}{\sin 10}-4\sin \left(90-20\right)\left[\mathbf{as}\mathbf{,}\mathbf{}\mathbf{cosec}\mathbf{}\mathbf{\theta }\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{sin}\mathbf{}\mathbf{\theta }}\right] \\ =\frac{1}{\sin 10}-4\cos 20\left[\mathbf{as}\mathbf{,}\mathbf{}\mathbf{sin}\mathbf{}\left(\mathbf{90}\mathbf{-}\mathbf{\theta }\right)\mathbf{}\mathbf{=}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{\theta }\right] \\ =\frac{1-4\cos 20.\sin 10}{\sin 10} \\ =\frac{1-2.2\cos 20.\sin 10}{\sin 10} \\ =\frac{1-2\left[\sin \left(20+10\right)-\sin \left(20-10\right)\right]}{\sin 10}\left[\mathbf{as}\mathbf{,}\mathbf{}\mathbf{2}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{A}\mathbf{}\mathbf{.}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{sin}\left(\mathbf{A}\mathbf{+}\mathbf{B}\right)\mathbf{-}\mathbf{sin}\mathbf{}\left(\mathbf{A}\mathbf{-}\mathbf{B}\right)\right] \\ =\frac{1-2\left[\sin 30-\sin 10\right]}{\sin 10} \\ =\frac{1-2\left({\displaystyle \frac{1}{2}-\sin 10}\right)}{\sin 10} \\ =\frac{1-1+2\sin 10}{\sin 10} \\ =2 \end{gathered}$$