Question

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

Answer: 1

Answer 1

`\bf \cfrac{cos(\theta )}{sin(\theta )}-\cfrac{cot(\theta )}{tan(\theta )}=1\implies \cfrac{cos(\theta )}{sin(\theta )}-\cfrac{(cos(\theta ))/(sin(\theta ))}{(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 )\\e 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.