Question

Prove that : tan²A + cot²A + 2 = sec²A. cosec²A

Answer 1

Answer:

tan²A + cot²A = sec²A. cosec²A -2

Answer 2

$\huge\bf\orange{Solution:-}$

tan²A + cot²A + 2 = sec²A × cosec²A

tan²A + cot²A = sec²A × cosec²A - 2

LHS = tan²A + cot²A

= (sec²A - 1 ) + (cosec²A - 1)

= sec²A + cosec²A - 2

$=\frac{1}{{cos}^{2}\alpha} + \frac{1}{{sin}^{2}\alpha} - 2 \\ \\ = \frac{{cos}^{2}\alpha + {sin}^{2}\alpha}{{cos}^{2}\alpha× {sin}^{2}\alpha}- 2 \\ \\ \huge\boxed{{sin}^{2}\theta + {cos}^{2}\theta = 1} \\ \\ = \frac{1}{{cos}^{2}\alpha\: {cosec}^{2}\alpha} -2\\ \\= {sec}^{2}\alpha \:{cosec}^{2}\alpha - 2$

HENCE PROVED