Question

Prove that. (1-tanA)²+(1-cotA)²=(sec A - Cosec A)²​

Answer 1

Step-by-step explanation:

L.H.S = (1- tanA)² + (1-cotA)²

= 1 - 2tanA + tan²A + 1 - 2cotA + cot²A

= sec²A - 2 tanA - 2cotA + cosec²A

= sec²A - 2( tanA - cotA) + cosec²A

= sec²A -2( tanA - 1/ tanA) + cosec²A

= sec²A - 2 (tan²A - 1)/tanA + cosec²A

= sec²A - 2( -sec²A) cosA/sinA + cosec²A

$= {sec}^{2} a + \frac{1}{ {cos}^{2}a } \frac{cos \: a}{sin \: a} + {cosec}^{2}a$

$= {sec}^{2} a + \frac{1}{cos \: a} \frac{1}{sin \: a} + {cosec}^{2} a$

= sec²A + secA cosecA + cosec²A

= (secA + cosecA)²

= R.H.S

hence proved.

I hope this will help you.

Thank you.

Happy to help.