If $5sin\theta=4$, prove that $\frac{1}{cos\theta}+\frac{1}{cot\theta}=3$.

Given: $5sin\theta=4$.

To do: To prove that $\frac{1}{cos\theta}+\frac{1}{cot\theta}=3$.

Solution: 

As given $5sin\theta=4$

$\Rightarrow sin\theta=\frac{4}{5}$

$\Rightarrow sin^{2}\theta=( \frac{4}{5})^{2}=\frac{16}{25}$

$\Rightarrow cos^{2}\theta=1-sin^{2}\theta=1-\frac{16}{25}=\frac{25-16}{25}=\frac{9}{25}$

$\Rightarrow cos\theta=\sqrt{\frac{9}{25}}=\frac{3}{5}$

$\Rightarrow cot\theta=\frac{cos\theta}{sin\theta}=\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3}{4}$

$\Rightarrow \frac{1}{cos\theta}+\frac{1}{cot\theta}=\frac{1}{\frac{3}{5}}+\frac{1}{\frac{3}{4}}=\frac{5}{3}+\frac{4}{3}=\frac{5+4}{3}=\frac{9}{3}=3$

Thus, it has been proved that $\frac{1}{cos\theta}+\frac{1}{cot\theta}=3$.

Tutorialspoint

Updated on: 10-Oct-2022

87 Views

Kickstart Your Career

Get certified by completing the course

Get Started