Question

(sin4theta-cos4thetha+1)cosec2thetha =2​

Answer 1

$\huge{\mathbb{\red{ANSWER:-}}}$

$\sf{(Sin^{4} O - Cos^{4} O + 1)(Cosec^{2} O)}$

$\sf{[(Sin^{4} O - Cos^{4} O) + 1](Cosec^{2} O)}$

$\sf\small\boxed{(a^{2} - b^{2}) = (a + b)(a - b)}$

$\sf{[(Sin^{2} O)^{2} - (Cos^{2} O)^{2} + 1](Cosec^{2} O)}$

$\sf{[(Sin^{2} O + Cos^{2} O)(Sin^{2} O - Cos^{2} O) + 1](Cosec^{2} O)}$

$\sf{\sf\boxed{Sin^{2} O + Cos^{2} O = 1}}$

$\sf{[1(Sin^{2} O - Cos^{2} O) + 1](Cosec^{2} O)}$

$\sf{[Sin^{2} O - Cos^{2} O + 1](Cosec^{2} O)}$

$\sf{[Sin^{2} O + 1 - Cos^{2} O](Cosec^{2} O)}$

$\sf{[Sin^{2} O + Sin^{2} O](Cosec^{2} O)}$

$\sf{2Sin^{2} O\times Cosec^{2} O}$

$\sf{2(SinO\times CosecO)^{2}}$

$\sf{\sf\boxed{SinO\times CosecO = 1}}$

$\sf{2(1)^{2}}$

$\sf{2(1)}$

$\sf{2}$

$\sf{HENCE \: PROOF}$

Extra Related Formulas :-

★ $\sf{(CosO\times SecO) = 1}$

★ $\sf{(tanO\times CotO) = 1}$

★ $\sf{[Sec^{2} O = (1 + tan^{2} O)]}$

★ $\sf{[Cosec^{2} O = (1 + Cot^{2} O)]}$

Answer 2

Question

( Sin4theta - cos4theta + 1 ) cosec2theta = 2

$\sf \color {purple} \large \huge \:ANSWER$

$( { \sin}^{4} 0 - { \cos}^{4} 0 + 1) { \cosec}^{2} 0$

$\sf \implies \: { (\sin}^{2} 0 {)}^{2} - ({ \cos}^{2} 0 {)}^{2} + 1 \: {cosec}^{2} 0.$

$\\ \sf \: \implies { ((\sin}^{2}) 0 + {( \cos }^{2}0 {)}^{2} \: ( { \sin}^{2} 0 - { \cos}^{2} 0 + 1) { \cosec}^{2} 0 + 1$

$\\ \implies \: ( { \sin}^{2} 0 - { \cos }^{2} 0 + 1) \: \dfrac{1}{ { \sin}^{2}0}$

As we know that

${ \sin}^{2} 0 + { \cos}^{2} = 1 \\ 1 = { \cos}^{2} 0 = { \sin}^{2}0$

$\\ \color{blue} \: \sf \implies(2 { \sin}^{2}0) \: \frac{1}{ { \sin}^{2}0 } = 2$