Question

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Q. no. 6​

Answer 1

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Answer 2

Solution:-

$\rm i) \implies \: \dfrac{ \sin {}^{2} \theta }{ \cos \theta} + \cos \theta = \sec \theta$

Now take lcm

$\rm \implies \: \dfrac{ \sin {}^{2} \theta + ( \cos \theta ) \cos \theta}{ \cos \theta }$

$\rm \implies \: \dfrac{ \sin {}^{2} \theta + \cos {}^{2} \theta}{ \cos \theta}$

We know that

$\rm \implies \: \sin {}^{2} x + \cos {}^{2} x = 1$

We get

$\rm \implies \: \dfrac{1}{ \cos \theta}$

We know that

$\rm \implies \: \dfrac{1}{ \cos x} = \sec x$

We get

$\rm \implies \: \dfrac{1}{ \cos \theta} = \sec \theta$

Hence proved

$\rm \: ii) \implies \: \cos {}^{2} \theta(1 + \tan^{2} \theta) = 1$

We know that

$\rm \implies \tan x = \dfrac{ \sin x}{ \cos x}$

Using this identities

$\rm \implies \cos {}^{2} \theta \bigg(1 + \dfrac{ \sin {}^{2} \theta }{ \cos {}^{2} \theta} \bigg)$

Now Take lcm

$\rm \implies \cos {}^{2} \theta \bigg( \dfrac{ \cos {}^{2} \theta + \sin {}^{2} \theta}{ \cos {}^{2} \theta } \bigg)$

$\rm \implies \cancel{ \cos {}^{2} \theta} \bigg( \dfrac{ \cos {}^{2} \theta + \sin {}^{2} \theta}{ \cancel{\cos {}^{2} \theta }} \bigg)$

We get

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

Hence proved