Question

(Sin a+cos a)(tan a+cot a)= sec a+ cosec a

Answer 1

Given:

$\bf (sin \: a + cos \: a)(tan \: a + cot \: a) = sec \: a + cosec \: a$

To prove :

• RHS = LHS

Proof :

Let's take LHS :-

$\sf \implies(sin \: a + cos \: a)(tan \: a + cot \: a)$

We know that :-

$\underline{ \boxed{ \bf \large \: tan \: x = \dfrac{sin \: x }{cos \: x}}}$

$\underline{ \boxed{ \bf \large \:co t \: x = \dfrac{cos\: x }{sin \: x}}}$

$\sf \implies(sin \: a + cos \: a) \bigg( \dfrac{sin \: a }{cos \: a} + \dfrac{ cos\: a }{ sin\: a} \bigg)$

$\sf \implies(sin \: a + cos \: a) \bigg( \dfrac{ {sin}^{2} a+ {cos}^{2}a }{ cos\: a \: sin\: a} \bigg)$

We know that :-

$\underline{ \boxed{ \bf \large {sin}^{2} x+ {cos}^{2}x = 1}}$

$\sf \implies(sin \: a + cos \: a) \bigg( \dfrac{ 1 }{ cos\: a \: sin\: a} \bigg)$

$\sf \implies \bigg( \dfrac{ sin \: a}{ cos\: a \: sin\: a} \bigg) + \bigg( \dfrac{ cos\: a}{ cos\: a \: sin\: a} \bigg)$

$\sf \implies \bigg( \dfrac{ \cancel {sin \: a}}{ cos\: a \: \: \cancel {sin \: a} } \bigg) + \bigg( \dfrac{ \cancel {cos\: a}}{ \cancel {cos\: a} \: \: sin\: a} \bigg)$

$\sf \implies \bigg( \dfrac{1}{ cos\: a } \bigg) + \bigg( \dfrac{ 1}{ sin\: a} \bigg)$

We know that :-

$\underline{ \boxed{ \bf \large \dfrac{1 }{cos \: x} = sec \: x} } \\ \underline{ \boxed{ \bf \large \dfrac{1 }{sin \: x} = cosec \: x} }$

$\sf \implies sec \: a + cosec \: a$

Therefore LHS = RHS

hence proved

Answer 2

Step-by-step explanation:

Taking L.H.S!!

(sin a + cos a)(tan a + cot a)

sin a(tan a + cot a) + cos a(tan a + cot a)

★ Formula:-

» cot ø = cos ø ÷ sin ø

» tan ø = sin ø ÷ cos ø

$\sin a ( \frac{ \sin \: a }{ \cos \: a } + \frac{ \cos \: a }{ \sin \: a} ) + \cos \: a( \frac{ \sin \: a}{ \cos \: a} + \frac{ \cos \: a }{ \sin \: a } )$

$sin \: a ( \frac{ {sin}^{2} a + { \cos }^{2} a}{ \sin \: a \: .\cos \: a } ) +cos \: a ( \frac{ {sin}^{2} a + { \cos }^{2} a}{ \sin \: a \: .\cos \: a } )$

★ Formula:-

» sin² ø + cos ² ø = 1

$sin \: a ( \frac{1}{ \sin \: a \: .\cos \: a } ) \: cos \: a ( \frac{1}{ \sin \: a \: .\cos \: a } )$

$sin \: a( \frac{1}{sin \: a} ).sin \: a( \frac{1}{cos \: a} ) + cos \: a( \frac{1}{sin \: a} ).cos \: a( \frac{1}{cos \: a} ) +$

$1.( \frac{sin \: a}{cos \: a} ) + ( \frac{cos \: a}{sin \: a} ) .1$

$( \frac{ {sin}^{2} a + {cos}^{2} a}{cos \: a.sin \: a} )$

$( \frac{ 1}{cos \: a.sin \: a} )$

$( \frac{1}{cos \: a} ).( \frac{1}{sin \: a} )$

★ Formula:-

» 1 ÷ cos ø = sec ø

» 1 ÷ sin ø = cosec ø

$sec \: a \: cosec \: a$

Hence Proved!!

L.H.S = R.H.S

hope you understand

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