Trignometry solve it fast

1.	Trignometry solve it fast
Answer» Given Expression: ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ $\sf{\dfrac{1\:-\:SinA}{1\:+\:SinA}\:=\:(SecA\:-\:<klux>TANA</klux>)^2}$ ⠀⠀⠀⠀⠀⠀⠀ SoluTion: ⠀⠀⠀⠀⠀⠀⠀ Taking RHS, ⠀⠀⠀⠀⠀⠀⠀ ★ CHANGING TanA and SecA in Sin and Cos. ⠀⠀⠀⠀⠀⠀⠀ $\longrightarrow$ $\sf{(\dfrac{1}{CosA}\:-\:\dfrac{SinA}{CosA})^2}$ ⠀⠀⠀⠀⠀⠀⠀ Taking LCM, ⠀⠀⠀⠀⠀⠀⠀ $\longrightarrow$ $\sf{(\dfrac{1\:-\:SinA}{CosA})^2}$ ⠀⠀⠀⠀⠀⠀⠀ Separating the SQUARES, ⠀⠀⠀⠀⠀⠀⠀ $\longrightarrow$ $\sf{\dfrac{(1\:-\:SinA)^2}{Cos^{2} A}}$ ⠀⠀⠀⠀⠀⠀⠀ We know that, ⠀⠀⠀⠀⠀⠀⠀ $\large{\boxed{\rm{\green{Cos^{2}\:A\:=\:1\:-\:Sin^2\:A}}}}$ ⠀⠀⠀⠀⠀⠀⠀ $\longrightarrow$ $\sf{\dfrac{(1\:-\:SinA)(1\:-\:SinA)}{1\:-\:Sin^2\:A}}$ ⠀⠀⠀⠀⠀⠀⠀ We also know that, ⠀⠀⠀⠀⠀⠀⠀ $\large{\boxed{\rm{\red{(\:a^2\:-\:b^2\:)\:=\:(a\:+\:b)(a\:-\:b)}}}}$ ⠀⠀⠀⠀⠀⠀⠀ $\longrightarrow$ $\sf{\dfrac{(1\:-\:SinA)(1\:-\:SinA)}{(1\:+\:SinA)(1\:-\:SinA)}}$ ⠀⠀⠀⠀⠀⠀⠀ $\longrightarrow$ $\sf{\dfrac{1\:-\:SinA}{1\:+\:SinA}}$ ⠀⠀⠀⠀⠀⠀⠀ $\longrightarrow$ LHS ⠀⠀⠀⠀⠀⠀⠀ Hence PROVED!

Answer»

Given Expression:

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\sf{\dfrac{1\:-\:SinA}{1\:+\:SinA}\:=\:(SecA\:-\:<klux>TANA</klux>)^2}$

⠀⠀⠀⠀⠀⠀⠀

SoluTion:

⠀⠀⠀⠀⠀⠀⠀

Taking RHS,

⠀⠀⠀⠀⠀⠀⠀

★ CHANGING TanA and SecA in Sin and Cos.

⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow$ $\sf{(\dfrac{1}{CosA}\:-\:\dfrac{SinA}{CosA})^2}$

⠀⠀⠀⠀⠀⠀⠀

Taking LCM,

⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow$ $\sf{(\dfrac{1\:-\:SinA}{CosA})^2}$

⠀⠀⠀⠀⠀⠀⠀

Separating the SQUARES,

⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow$ $\sf{\dfrac{(1\:-\:SinA)^2}{Cos^{2} A}}$

⠀⠀⠀⠀⠀⠀⠀

We know that,

⠀⠀⠀⠀⠀⠀⠀

$\large{\boxed{\rm{\green{Cos^{2}\:A\:=\:1\:-\:Sin^2\:A}}}}$

⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow$ $\sf{\dfrac{(1\:-\:SinA)(1\:-\:SinA)}{1\:-\:Sin^2\:A}}$

⠀⠀⠀⠀⠀⠀⠀

We also know that,

⠀⠀⠀⠀⠀⠀⠀

$\large{\boxed{\rm{\red{(\:a^2\:-\:b^2\:)\:=\:(a\:+\:b)(a\:-\:b)}}}}$

⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow$ $\sf{\dfrac{(1\:-\:SinA)(1\:-\:SinA)}{(1\:+\:SinA)(1\:-\:SinA)}}$

⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow$ $\sf{\dfrac{1\:-\:SinA}{1\:+\:SinA}}$

⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow$ LHS

⠀⠀⠀⠀⠀⠀⠀