Question

༒ Bʀᴀɪɴʟʏ Sᴛᴀʀs

༒Mᴏᴅᴇʀᴀᴛᴇʀs

༒Other best user

Prove that.

please give me right answer
all are spamming
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Answer 1

⠀⠀⠀⌬⠀Prove That :

$\qquad \bigstar \:\: \pmb{\sf \sqrt{\dfrac{ 1 - Cos \:\big( \alpha \big) }{ 1 + Cos \big( \alpha \:\big) } }\:\:=\:\: \dfrac{ sin \:\big( \alpha \big) }{ 1 + cos \: \big( \alpha \big)}\:}\:$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

$\qquad \dashrightarrow \sf \sqrt{\dfrac{ 1 - Cos \:\big( \alpha \big) }{ 1 + Cos \big( \alpha \:\big) } }\:\:=\:\: \dfrac{ sin \:\big( \alpha \big) }{ 1 + cos \: \big( \alpha \big)\:} \:$

$\qquad \bigstar \:\:\underline {\pmb{\sf{\purple { By \:\:Taking \:\:L.H.S \:\::\:}}}}$

$\qquad \dashrightarrow \sf \sqrt{\dfrac{ 1 - Cos \:\big( \alpha \big) }{ 1 + Cos \big( \alpha \:\big) } }\:\: \:$

$\qquad \dashrightarrow \sf \sqrt{\dfrac{ 1 - Cos \:\big( \alpha \big) }{ 1 + Cos \big( \alpha \:\big) }} \:\times \: \sqrt{\dfrac{ 1 - Cos \:\big( \alpha \big) }{ 1 + Cos \big( \alpha \:\big) }} \:\: \:$

$\qquad \dashrightarrow \sf \sqrt{\dfrac{\Big\{ 1 - Cos \:\big( \alpha \big)\:\Big\} \Big\{ 1 - Cos \big( \alpha \:\big) \Big\} }{ 1 + Cos \big( \alpha \:\big)\Big\{ 1 - Cos \big( \alpha \:\big) \Big\} } }\:\: \:$

$\qquad \dashrightarrow \sf \sqrt{\dfrac{ 1^2 - Cos^2 \:\big( \alpha \big) }{ \Big\{ 1 + Cos \big( \alpha \:\big) \Big\}^2 }} \:\:\qquad \:\because \:\bigg\lgroup \sf{ ( a^2 - b^2 )\:=\: ( a + b ) \:( a - b) }\bigg\rgroup\:$

$\qquad \dashrightarrow \sf \sqrt{\dfrac{ 1 - Cos^2 \:\big( \alpha \big) }{ \Big\{ 1 + Cos \big( \alpha \:\big) \Big\}^2 } }\:\: \:$

$\qquad \dashrightarrow \sf \sqrt{\dfrac{ Sin^2 \:\big( \alpha \big) }{ \Big\{ 1 + Cos \big( \alpha \:\big) \Big\}^2 } }\:\: \qquad \:\because \:\bigg\lgroup \sf{ \:1\:-\:Cos^2\theta \: \:=\:sin^2\:\theta }\bigg\rgroup \:$

$\qquad \dashrightarrow \sf \dfrac{ Sin \:\big( \alpha \big) }{ 1 + Cos \big( \alpha \:\big) } \:\: \:$

$\qquad \dashrightarrow \:\: \purple {\pmb{\sf \sqrt{\dfrac{ 1 - Cos \:\big( \alpha \big) }{ 1 + Cos \big( \alpha \:\big) } }\:\:=\:\: \dfrac{ sin \:\big( \alpha \big) }{ 1 + cos \: \big( \alpha \big)}\:}}\:\:\bigstar$

$\qquad \therefore \:\:\underline {\pmb{ \bf \: Hence \:,\:Verified \:\:!\:}}\:$