Question

Prove that tan²∅ + cot²∅ + 2 = sec²∅ - cosec²∅..

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Answer 1

Step-by-step explanation:

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Answer 2

CORRECT QUESTION :–

$\\ \: \: { \bold{ prove \: \: that \: : \: \tan {}^{2} ( \theta) + { \cot}^{2} ( \theta) + 2 = { \sec}^{2} ( \theta) . {cosec}^{2} ( \theta)}} \: \:$

TO PROVE :–

$\\ \: \: { \bold{ \tan {}^{2} ( \theta) + { \cot}^{2} ( \theta) + 2 = { \sec}^{2} ( \theta) . {cosec}^{2} ( \theta)}} \: \:$

SOLUTION :–

• Let's take L.H.S. –

$\\ \: \: { \bold{ = \tan {}^{2} ( \theta) + { \cot}^{2} ( \theta) + 2 }} \: \:$

• We should write this –

$\\ \: \: { \bold{ = \tan {}^{2} ( \theta) + { \cot}^{2} ( \theta) + 2(1) }} \: \:$

• We know that –

$\\ \: \: \longrightarrow { \bold{ \tan ( \theta) = \dfrac{1}{ { \cot} ( \theta) }}} \: \:$

$\\ \: \: \longrightarrow { \bold{ \tan ( \theta) \times{ \cot} ( \theta) = 1}} \: \:$

• So that –

$\\ \: \: { \bold{ = \tan {}^{2} ( \theta) + { \cot}^{2} ( \theta) + 2\tan( \theta). \cot( \theta) }} \: \:$

• Using identity –

$\\ \: \: \longrightarrow \: \: { \bold{ {(a + b)}^{2} = {a}^{2} + { b}^{2} + 2ab }} \: \:$

$\\ \: \: { \bold{ = [\tan ( \theta) + { \cot}( \theta) ] {}^{2} }} \: \:$

• We know that –

$\\ \: \: \longrightarrow { \bold{ \tan ( \theta) = \dfrac{ \sin( \theta) }{ { \cos} ( \theta) }}} \: \:$

$\\ \: \: \longrightarrow { \bold{ \cot ( \theta) = \dfrac{ \cos( \theta) }{ { \sin} ( \theta) }}} \: \:$

• So that –

$\\ \: \: { \bold{ = \left[ \dfrac{ \sin( \theta) }{ { \cos} ( \theta) } + \dfrac{ \cos( \theta) }{ { \sin} ( \theta) } \right]^{2} }} \: \:$

$\\ \: \: { \bold{ = \left[ \dfrac{ \sin {}^{2} ( \theta) + { \cos}^{2}( \theta) }{ { \cos} ( \theta) . \sin( \theta) } \right]^{2} }} \: \:$

$\\ \: \: { \bold{ = \left[ \dfrac{ 1 }{ { \cos} ( \theta) . \sin( \theta) } \right]^{2} }} \: \: \: \: \: \:[ \: \: \because \: \: \sin {}^{2} ( \theta) + { \cos}^{2}( \theta) = 1] \:$

$\\ \: \: { \bold{ = \left[ \sec( \theta) .cosec( \theta) \right]^{2} \: \: \: \: \: \: \left[ \: \: \because \: \: \sec ( \theta) = \dfrac{1}{ \cos( \theta)} \: \: and \: \: { \cosec}( \theta) = \dfrac{1}{ \sin( \theta) } \right] \: }}$

$\\ \: \: { \bold{ = \sec {}^{2} ( \theta) .cosec {}^{2} ( \theta) \: \: \: }}$

$\\ \: \: { \bold{ = R.H.S. \: \: \: \: \: \: (Hence \: \: proved) }}$