Question

If $cos\theta =\frac{3}{5}$, show that $\frac{(sin\theta -cot\theta )}{2tan\theta }=\frac{3}{160}$

Answer 1

$cos\theta =\frac{3}{5}$

But $cos\theta =\frac{Adjacentside}{Hypotenuseside}$

So Hypotenuse = 5 and Adjacent side = 3

According to Pythagoras theorem,

$(\text{Hypotenuse}{)}^2=(\text{Opposite side}{)}^2+\left(\text{Adjacent side}{)}^2\right(\text{Opposite side}{)}^2=(\text{Hypotenuse}{)}^2-(\text{Adjacent side}{)}^2(\text{Opposite side}{)}^2=5^2-3^2=25-9=16\text{Opposite side}=\sqrt{16}=4$

$sin\theta =\frac{Oppositeside}{Hypotenuseside}=\frac{4}{5}$

$tan\theta =\frac{Oppositeside}{Adjacentside}=\frac{4}{3}$

$cot\theta =\frac{Adjacentside}{Oppositeside}=\frac{3}{4}$

$\frac{sin\theta -cot\theta }{2tan\theta }=\frac{\frac{4}{5}-\frac{3}{4}}{2\times \frac{4}{3}}=\frac{\frac{16-15}{20}}{\frac{8}{3}}=\frac{1}{20}\times \frac{8}{3}=\frac{3}{160}$