Question

if 5 cot θ = 4, then find the value of (2 sin²θ + 3 cos²θ) / (7 sin θ . cos θ).

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{can}\mathrm{simplify}, \\ \frac{2{\sin }^2\mathrm{\theta }+3{\cos }^2\mathrm{\theta }}{7\mathrm{cos\theta sin\theta }},\mathrm{Now}\mathrm{divide}\mathrm{by}{\sin }^2\mathrm{\theta }\mathrm{to}\mathrm{both}\mathrm{the}\mathrm{denominator}\mathrm{and}\mathrm{numerator},\mathrm{we}\mathrm{get} \\ =\frac{{\displaystyle \frac{2{\sin }^2\mathrm{\theta }+3{\cos }^2\mathrm{\theta }}{{\sin }^2\mathrm{\theta }}}}{{\displaystyle \frac{7\mathrm{cos\theta sin\theta }}{{\sin }^2\mathrm{\theta }}}}=\frac{2+3{\cot }^2\mathrm{\theta }}{7\mathrm{cot\theta }} \\ \mathrm{Now}\mathrm{cot\theta }\mathrm{is}\mathrm{given}\mathrm{as}\frac{4}{5},\mathrm{Putting}\mathrm{this}\mathrm{we}\mathrm{get} \\ =\frac{2+3\times {\displaystyle \frac{16}{25}}}{7\times \frac{4}{5}}=\frac{{\displaystyle \frac{50+48}{25}}}{{\displaystyle \frac{28}{5}}}=\frac{98}{140}=\frac{49}{70} \end{gathered}$$.