Question

Mathematics question (Optional math) ​

Answer 1

`\large{\pink{\underline\textsf{To Prove :-}}}`

`\sqrt{ (1 + \sin \theta )/(1 - \sin \theta ) } = \sec \theta + \tan \theta`

`\huge{\orange{\underline{\textsf{Solution :-}}}}`

`\sf \sqrt{ (1 + \sin \theta )/(1 - \sin \theta ) }`

• First we have to rationalize it to remove the square root
• Now we will multiply √1+sinθ both sides

`\sqrt{ (1 + \sin \theta )/(1 - \sin \theta ) } * \sqrt{ (1 + \sin \theta )/(1 - \sin \theta ) } \\ \sqrt{ \frac{ {(1 + \sin \theta) }^(2) }{(1 + \sin \theta)(1 - \sin \theta) } } \\ \sqrt{ \frac{ {(1 + \sin \theta) }^(2) }{1 - { { \sin}^(2) \theta } } }`

• We know that sin²θ + cos²θ = 1
• so cos²θ = 1 - sin²θ

`\sqrt{ \frac{( {1 + \sin \theta })^(2) }{ { \cos}^(2) \theta} } \\ (1 + \sin \theta )/( \cos \theta ) \\ (1)/( \cos \theta) + ( \sin \theta )/( \cos \theta )`

`\sf{\red { \sec \theta + \tan \theta}}`

`\sf \huge \purple {HENCE \: \: PROVED! }`

Answer 2

Hi mate!

To prove:-

`\bf \sqrt{ (1 + \sin( \theta))/(1 - \sin( \theta)) } = \sec( \theta) - \tan( \theta)`

Explanation:

By taking LHS,

`\maltese \: \: \large \bf \sqrt{ (1 + \sin( \theta))/(1 - \sin( \theta)) }`

By rationalisation method,

`: \longrightarrow \bf \sqrt{ (1 + \sin( \theta))/(1 - \sin( \theta)) * (1 + \sin( \theta))/(1 + \sin( \theta)) } \\ \\ : \longrightarrow \frac{ {( √(1 + \sin\theta )) }^(2) }{( √(1 - \sin( \theta)))(√(1 + \sin( \theta))) } \\ \\ : \longrightarrow \frac{ {( √(1 + \sin\theta )) }^(2) }{ \sqrt{ {1}^(2) - { \sin}^(2) \theta } } \: \\ \\ : \longrightarrow \: (1 + \sin \theta)/( \cos \theta ) \\ \\ \longrightarrow \: (1)/( \cos \theta ) + ( \sin \theta)/( \cos \theta)`

`\: \: \: \: \: \bigg\lgroup \because \: (1)/( \cos \theta )\: = \sec \theta \: \\ \: \: \: \: \: \: \: \: \: \: \: ( \sin \theta)/( \cos \theta) = \tan \theta \bigg \rgroup`

`\implies \boxed{\pink{ \sec \theta + \tan \theta }}`

Now taking RHS

`\sf \sec \theta + \tan \theta`

`\implies \: \: (1)/( \cos \theta ) + ( \sin \theta)/( \cos \theta) \\ \\ \implies (1 + \sin \theta )/( \cos \theta )`

Hence, proved