Question

sinA(1-tanA)-cos(1-cotA)= cosecA - secA​

Answer 1

Step-by-step explanation:

$sinA(1 - tanA) - cosA(1 - cotA) \\ = sinA(1 - \frac{sinA}{cosA} ) - cosA(1 - \frac{cosA}{sinA} ) \\ = \frac{sinA(cosA - sinA)}{cosA} - \frac{cosA(sinA - cosA)}{sinA} \\ = \frac{sinAcosA - {sin}^{2}A }{cosA} - \frac{sinAcosA - {cos}^{2} A}{sinA} \\ = \frac{ {sin}^{2}AcosA - {sin}^{3} A - sinA {cos}^{2} A + {cos}^{3}A }{sinAcosA} \\ = \frac{(1 - {cos}^{2}A)(cosA) - {sin}^{3} A - (sin A)(1 - {sin}^{2} A) + {cos}^{3} A}{sinAcosA} \\ = \frac{cosA - {cos}^{3}A - {sin}^{3} A - sinA + {sin}^{3}A + {cos}^{3} A}{sinAcosA} \\ = \frac{cosA - sinA}{sinAcosA} \\ = \frac{cosA}{sinAcosA} - \frac{sinA}{sinAcosA} \\ = \frac{1}{sinA} - \frac{1}{cosA} = cosecA - secA$