Question

(sec A- cos A ) ( cot A + tan A ) = tan A.sec ​

Answer 1

(sec A - cos A ) ( cot A + tan A )

= (1/cosA - cos A) (cosA/sinA + sinA/cosA)

= (1-cos²A)/cosA × (cos²A+sin²A)/(sinA.cosA)

= sin²A/cosA × 1/sinA.cosA

= sinA/cosA × 1/cosA

= tan A . sec A

Answer 2

$\large{\underline{\underline{ \bf{⎆Question:-}}}}$

$\small\fbox\red{To \: prove : - }$

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✰ (sec A- cos A ) ( cot A + tanA ) = tan A.sec A

$\large{\underline{\underline{ \bf{☂ Solution:-}}}}$

$\small\fbox\red{ Proof : - }$

$\longmapsto\tt{(secA-cosA)(cotA+tanA) }$

• Assume angle 'A' as 'a'

$\longmapsto\tt{( \frac{1}{cos \: a} - cos \: a)( \frac{cos \: a}{sin \: a} + \frac{sin \: a \: }{cos \: a \: } }$

$\longmapsto\tt{( \frac{1 \: - \: {cos}^{2} a}{cos \: a \: } )( \frac{ {cos}^{2}a + {sin}^{2}a }{sin \: a \: .cos \: a} ) }$

$\huge{\underbrace{\overbrace{\color{red}{Formula \:Used}}}}$

• 1 - cos²a = sin²a
• cos²a + sin²a = 1

$\longmapsto\tt{( \frac{ {sin}^{2}a }{cos \: a \: } )( \frac{ 1 }{sin \: a \: .cos \: a} ) }$

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

$\longmapsto\tt{( \frac{(\cancel{sina}) \times sin \: a}{cos \: a \: } )( \frac{ 1 }{(\cancel{sina}) .cos \: a} ) }$

$\longmapsto\tt{( \frac{ sin \: a}{cos \: a \: } ) \times ( \frac{ 1 }{cos \: a} ) }$

$\longmapsto\tt{tan \: a \times sec \: a }$

$\underline{ \boxed{ \sf{ \longmapsto\tt{tan \: a \: sec \: a } }}}$

$\small\fbox\red{ Proved .}$