Question

Prove that
cos/1-tan + sin/1-cot = sin + cos​

Answer 1

༄༄Answer༄༄

$\tt\huge{\green{\underline{\underline{Given :-}} }}$

${\orange{\frac{cos}{1-tan }+\frac{ sin}{1-cot} = sin + cos}}$

$\tt\huge{\blue{\underline{\underline{By \:L.H.S}}} }$

$⟹\green{\huge{\frac{cosA}{1-(\frac{sinA}{cosA}) }+ \frac{sinA}{1-(\frac{cosA}{sinA})}}}$

$⟹\pink{ \huge{\frac{cos²A}{(cosA-sinA)}-\frac{sin²A}{sinA-cosA}}}$

$⟹ \purple{\huge{\frac{cos²A}{cosA-sinA} - \frac{sin²A}{cosA-sinA}}}$

$⟹ \orange{\:\:\:\huge{\frac{cos²A-sin²A}{cos²A-sin²A}}}$

$⟹ \red{\huge{\frac{(cosA+sinA)(cosA-sinA)}{cosA-sinA}}}$

$⟹ \blue{ \tt{\boxed{\green {\:\:\: (sinA+cosA)}}}}$

Answer 2

$\tt\huge{\green{\underline{\underline{Given :-}} }}$

${\orange{\frac{cos}{1-tan }+\frac{ sin}{1-cot} = sin + cos}}$

$\tt\huge{\blue{\underline{\underline{By \:L.H.S}}} }$

$⟹\green{\huge{\frac{cosA}{1-(\frac{sinA}{cosA}) }+ \frac{sinA}{1-(\frac{cosA}{sinA})}}}$

$⟹\pink{ \huge{\frac{cos²A}{(cosA-sinA)}-\frac{sin²A}{sinA-cosA}}}$

$⟹ \purple{\huge{\frac{cos²A}{cosA-sinA} - \frac{sin²A}{cosA-sinA}}}$

$⟹ \orange{\:\:\:\huge{\frac{cos²A-sin²A}{cos²A-sin²A}}}$

$⟹ \red{\huge{\frac{(cosA+sinA)(cosA-sinA)}{cosA-sinA}}}$

$⟹ \blue{ \tt{\boxed{\green {\:\:\: (sinA+cosA)}}}}$