Question

please solve my question correctly correctly correctly....don't spam......explanation must...​

Answer 1

Solution :-

From given :-

tan²¢ = 1-e²

e²= 1-tan²¢ ___(1)

Now , we have to prove -sec¢ + tan³¢ cosec¢ ,= (2-e²)^³/2

Let's play with RHS .

=>(2-e²)^3/2

=>{2-{1-tan²¢)} ^³/2 ____from (1)

=> (1+tan²¢ ) ^3/2

=> (sec²¢)^3/2

=> sec³¢

Now let's play with LHS .

sec¢ + tan³¢ cosec¢

=> 1/cos¢ + sin³¢/cos³¢ *1/sin¢

=> 1/cos¢ + sin²¢ /cos³¢

=> cos²¢ + sin²¢ /cos³¢ (taking LCM}

=> 1/cos³¢ = sec³¢

LHS = RHS PROVED

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Hope it's helpful

Answer 2

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies {tan}^{2} \theta = 1 - {e}^{2} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies sec \theta + {tan}^{3} \theta \: cosec \theta = {(2 - {e}^{2} )}^{ \frac{3}{2} }$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies {tan}^{2} \theta = 1 - {e}^{2} \\ \\ \tt \circ \: {tan}^{2} \theta= {sec}^{2} \theta - 1 \\ \\ \tt: \implies {sec}^{2} \theta - 1 = 1 - {e}^{2} \\ \\ \tt: \implies {sec}^{2} = 2 - {e}^{2} - - - - - (1) \\ \\ \bold{To \:Prove: } \\ \tt: \implies sec \theta + {tan}^{3} \theta \: cosec \theta = ({1 - {e}^{2} })^{ \frac{3}{2} } \\ \\ \bold{Proof : } \\ \tt: \implies sec \theta + {tan}^{3} \theta \: cosec \theta \\ \\ \tt\circ\:tan\theta=\frac{sin\theta}{cos\theta}\\\\ \tt\circ\:cosec\theta=\frac{1}{sin\theta}\\ \\ \tt: \implies sec \theta + \frac{sin^{3} \theta}{ {cos}^{3} \theta } \times \frac{1}{sin \theta} \\ \\ \tt: \implies sec \theta + \frac{ {sin}^{2} \theta }{ {cos}^{3} \theta }$

$\tt: \implies \frac{1}{cos \theta} + \frac{ {sin}^{2} \theta}{ {cos}^{3} \theta}\\ \\ \tt: \implies \frac{ {cos}^{2} \theta + {sin}^{2} \theta}{ {cos}^{3} \theta} \\\\ \tt\circ\:sin^{2}\theta+cos^{2}\theta=1\\\\ \tt: \implies \frac{1}{ {cos}^{3} \theta } \\ \\ \tt\circ\:\frac{1}{cos\theta}=sec\theta \\\\ \tt: \implies {sec}^{3} \theta \\ \\ \tt: \implies sec \theta \times {sec}^{2} \theta \\ \\ \tt: \implies \sqrt{2 - {e}^{2} } \times ({2 - e}^{2} ) \\ \\ \tt: \implies ( {2- {e}^{2} })^{ \frac{1}{2} } \times {(2 - e}^{2} ) \\ \\ \text{Bases\:are\:same\:so\:power\:can\:be\:add} \\ \green{\tt: \implies {(2- {e}^{2}) }^{ \frac{3}{2} } } \\ \\ \: \: \: \: \: \: \: \: \: \huge\green{\bold{proved }}$