Question

Prove that (1+cota-coseca) (1+tana+seca) =2

Answer 1

Ello..✌✌

[1+cotA-CosecA]*[1+tanA+secA]

= 1 + tanA + secA + cotA + 1 + cosecA - (1/sinA) - (1/cosA) - secAcosecA

= 1 + tanA + secA + cotA + 1 + cosecA - cosecA - secA - secAcosecA

= 2 + tanA + cotA - secAcosecA

= 2 + [(sin^2A + cos^2A)/(sinAcosA)] - secAcosecA

= 2 + 1/(sinAcosA) - secAcosecA

= 2 + secAcosecA - secAcosecA

= 2

Hence, LHS = RHS.

Answer 2

$\sf \huge \underline{ \fbox{ \: To \: prove : \: \: }}$

$\bigg (1+ \cot{(a)}- \cosec{(a)} \bigg) \bigg(1+ \tan{(a)}+ \sec{(a)} \bigg) = 2$

$\sf \huge \underline{ \fbox{ \: Proof : \: \: }}$

LHS

$\sf \hookrightarrow \bigg (1+ \cot{(a)}- \cosec{(a)} \bigg) \bigg(1+ \tan{(a)}+ \sec{(a)} \bigg) \\ \\ \sf \hookrightarrow \bigg(1 + \frac{ \cos{(a)}}{ \sin{(a)}} - \frac{1}{ \sin{(a)}} \bigg ) \bigg( 1 + \frac{ \sin{(a)}}{ \cos{(a)}} + \frac{1}{ \cos{(a)}} \bigg) \\ \\ \sf \hookrightarrow \bigg( \frac{ \sin{(a)} + \ cos{(a) }- 1}{ \sin{(a)}} \bigg ) \bigg( \frac{ \cos{(a)} + \sin{(a) }+ 1}{ \cos{(a)}} \bigg) \\ \\ \sf \hookrightarrow \frac{ { \bigg( \sin{(a)} + \cos{(a)} \bigg)}^{2} - {(1)}^{2} }{ \sin{a } \times \cos{a}} \\ \\ \sf \hookrightarrow \frac{{ \sin}^{2}a + { \cos}^{2} a + 2 \sin{(a)} \cos{(a) }- 1 }{ \sin{(a)} \times \cos{(a)}} \\ \\ \sf \hookrightarrow \frac{1 + 2 \sin{(a)} \cos{(a)} - 1}{ \sin{(a)} \times \cos{(a)}} \\ \\ \sf \hookrightarrow \frac{2 \sin{(a)} \cos{(a)}}{ \sin{(a)} \cos{(a)}} \\ \\ \sf \hookrightarrow 2$

$\red \star$LHS = RHS

Hence proved