Question

(tan 0-cot0)² + 2= sec²0+cosec² 0 - 2​

Answer 1

Question:-

LHS :-

$\\ \bullet \quad { \bigg( \tan \theta - \cot \theta \bigg)}^{2} + 2$

RHS :-

$\\ \bullet \quad { \sec }^{2} \theta + { \cosec}^{2} \theta - 2$

$\\ \quad \dag \underline{ \sf Method \: 1 : - }$

◕ Solve by simplifying the LHS

$\\ \quad \longrightarrow \sf \bigg( { \tan} \theta + { \cot }\theta \bigg)^{2} + 2$

$\\ \quad \longrightarrow \sf { \tan}^{2} \theta + { \cot }^{2} \theta - 2 \tan \theta. \cot \theta + 2$

$\\ \quad \longrightarrow \sf { \tan}^{2} \theta + { \cot}^{2} \theta - 2 \times \tan \theta \times \frac{1}{ \tan \theta} + 2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bigg\{ \cot \theta = \frac{1}{ \tan \theta} \bigg\}$

$\\ \quad \longrightarrow \sf { \tan}^{2} \theta + { \cot}^{2} \theta$

$\\ \quad \longrightarrow \sf { \sec}^{2} \theta - 1 + { \cosec}^{2} \theta - 1 \: \: \: \: \: \: \: \: \: \: \: \bigg \{ { \tan}^{2} \theta = { \sec}^{2} \theta - 1 \bigg \} \bigg \{ { \cot}^{2} \theta = { \cosec}^{2} \theta - 1 \bigg \}$

$\\ \quad \longrightarrow \boxed{\boxed{\sf { \sec}^{2} \theta + { \cosec}^{2} \theta - 2 } }\bigstar$

$\\ \quad \dag \underline{ \sf Method \: 2 : - }$

◕ Solve by simplifying the RHS

$\\ \quad \longrightarrow \sf { \sec}^{2} \theta + { \cosec}^{2} \theta - 2$

$\\ \quad \longrightarrow \sf 1 + { \tan}^{2} \theta + 1 + { \cot}^{2} \theta - 2$

$\\ \quad \longrightarrow \sf { \tan}^{2} \theta + { \cot}^{2} \theta$

$\\ \quad \longrightarrow \sf { \bigg( \tan \theta - \cot \theta \bigg)}^{2} - 2 \times \tan \theta \times \cot \theta \: \: \: \: \: \: \: \: \: \: \: \bigg \{ {(a \pm b)}^{2} - 2ab = {a}^{2} + {b}^{2} \bigg \}$

$\\ \quad \longrightarrow \sf { \bigg( \tan \theta - \cot \theta \bigg)}^{2} - 2 \times \tan \theta \times \dfrac{1}{ \tan \theta}$

$\\ \quad \longrightarrow \boxed{ \boxed{\sf \bigg( \tan \theta - \cot \theta \bigg)^{2} - 2 }} \bigstar$

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