Question

Prove: csc theta/sin theta - cot theta/tan thetaAnswer: 1

Answer 1

$\bf \cfrac{cos(\theta )}{sin(\theta )}-\cfrac{cot(\theta )}{tan(\theta )}=1\implies \cfrac{cos(\theta )}{sin(\theta )}-\cfrac{\frac{cos(\theta )}{sin(\theta )}}{\frac{sin(\theta )}{cos(\theta )}}=1 \\\\\\ \cfrac{cos(\theta )}{sin(\theta )}-\cfrac{cos(\theta )}{sin(\theta )}\cdot \cfrac{cos(\theta )}{sin(\theta )}=1 \implies \cfrac{cos(\theta )}{sin(\theta )}-\cfrac{cos^2(\theta )}{sin^2(\theta )}=1$

$\bf \cfrac{cos(\theta )sin(\theta )~~-~~cos^2(\theta )}{sin^2(\theta )}=1\implies cos(\theta )sin(\theta )-cos^2(\theta )\ne sin^2(\theta )$


now, if we move the second fraction over, we'd get the equation of,

$\bf \cfrac{cos(\theta )}{sin(\theta )}-\cfrac{cot(\theta )}{tan(\theta )}=1\implies \cfrac{cos(\theta )}{sin(\theta )}=1+\cfrac{cot(\theta )}{tan(\theta )}$


now, check the picture below, they are definitely not equal to one another.