Question

prove trigonometry question

Answer 1

Divide lhs by cosx

Secx + tanx) /( secx - tanx)
( secx +tanx) ^2/ sec^2x - tan^2x
( secx +tanx) ^2 rhs ....... Proved

Answer 2

Theta is written as A ,




LHS = >

$\dfrac{ 1 + sinA }{ 1 - sinA } \\ \\ \\ \bold{By\:Rationalization ,} \\ \\ \\ \\ \dfrac{1 + sinA}{1 - sinA} \times \dfrac{1 + sinA}{1+sinA}$



$= > \dfrac{ ( 1 + sinA)^{2}}{1^{2}- sin^{2}A } \\ \\ \\ = > \frac{ 1^{2} + sin^{2}A + 2sinA }{1 - sin^{2}A} \\ \\ \\ = > \dfrac{ 1 + sin^{2}A + 2sinA}{ cos^2A}$





RHS = >



$= > ( secA + tanA)^{2} \\ \\ \\ = > sec^{2}A + tan^{2}A + 2secA.tanA \\ \\ \\ = >sec^{2}A + tan^{2}A + 2secA.tanA \\ \\ \\ = > \frac{1}{cos {}^{2}A } + \frac{sin {}^{2}A }{ {cos}^{2}A } + (2 \times \frac{1}{cosA} \times \frac{sin A}{ {cos}A } )$




$= > \frac{1 + sin {}^{2}A }{cos {}^{2} A} + 2( \frac{sinA}{ {cos}^{2}A } ) \\ \\ \\ = > \frac{1 + {sin}^{2} A + 2sinA}{ {cos}^{2} A }$








LHS = RHS


Hence, proved.