Question

(sin⁴A - cos⁴A + 1) cosec ² A = 2​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

1.

${sin}^{2} \theta \: + {cos}^{2} \theta \: = 1$

2.

$\displaystyle \: cosec \theta \: = \frac{1}{sin \theta \: }$

3.

${x}^{2} - {y}^{2} = (x + y)(x - y)$

TO PROVE

$( {sin}^{4} A - {cos}^{4} A + 1) \times {{cosec} }^{2} A = 2$

EVALUATION

$( {sin}^{4} A - {cos}^{4} A + 1) \times {{cosec} }^{2} A$

$= \{ \{ ( { {sin}^{2}A) }^{2} - ({{cos}^{2}A) }^{2} \} + 1 \} \ \times {{cosec} }^{2} A$

$= \{ ({sin}^{2} A + {cos}^{2} A)({sin}^{2} A - {cos}^{2} A) + 1 \} \times {cosec }^{2} \:A$

$= ({sin}^{2} A - {cos}^{2} A + 1 ) \times {cosec }^{2} \:A$

$= \displaystyle \: 2 {sin}^{2} A \times \frac{1}{{sin}^{2} \: A}$

$= 2$