Question

prove that (tanA+CosecB)²-(CotB-SecA)² = 2tanACotB(CosecA+SecB) ​

Answer 1

Answer:

ᴡᴇ ʜᴀᴠᴇ,

ʟʜꜱ = (ᴛᴀɴᴀ + ᴄᴏꜱᴇᴄ ʙ)2 - (ᴄᴏᴛʙ - ꜱᴇᴄ ᴀ)2

⇒ ʟʜꜱ = (ᴛᴀɴ2ᴀ + ᴄᴏꜱᴇᴄ2ʙ + 2ᴛᴀɴᴀ ᴄᴏꜱᴇᴄʙ ) - (ᴄᴏᴛ2ʙ + ꜱᴇᴄ2ᴀ - 2ᴄᴏᴛʙ ꜱᴇᴄᴀ)

⇒ ʟʜꜱ = (ᴛᴀɴ2ᴀ - ꜱᴇᴄ2ᴀ)+(ᴄᴏꜱᴇᴄ2ʙ - ᴄᴏᴛ2ʙ)+2ᴛᴀɴᴀ ᴄᴏꜱᴇᴄʙ +2ᴄᴏᴛʙ ꜱᴇᴄᴀ

ʙᴜᴛ, ꜱᴇᴄ2ᴀ - ᴛᴀɴ2ᴀ =1 & ᴄᴏꜱᴇᴄ2ᴀ - ᴄᴏᴛ2 ᴀ = 1

∴ ʟʜꜱ = -1 + 1 + 2 ᴛᴀɴᴀ ᴄᴏꜱᴇᴄʙ + 2ᴄᴏᴛʙ ꜱᴇᴄᴀ

⇒ ʟʜꜱ = 2 (ᴛᴀɴᴀ ᴄᴏꜱᴇᴄʙ + ᴄᴏᴛʙ ꜱᴇᴄᴀ)

⇒ ʟʜꜱ = 2 ᴛᴀɴᴀ ᴄᴏᴛʙ(ᴄᴏꜱᴇᴄʙᴄᴏᴛʙ+ꜱᴇᴄᴀᴛᴀɴᴀ) [ᴅɪᴠɪᴅɪɴɢ ᴀɴᴅ ᴍᴜʟᴛɪᴘʟʏɪɴɢ ʙʏ ᴛᴀɴᴀ ᴄᴏᴛʙ]

⇒ ʟʜꜱ = 2ᴛᴀɴ ᴀ ᴄᴏᴛʙ{1ꜱɪɴʙᴄᴏꜱʙꜱɪɴʙ+1ᴄᴏꜱᴀꜱɪɴᴀᴄᴏꜱᴀ} [ꜱɪɴᴄᴇ, ᴄᴏꜱᴇᴄᴀ.ꜱɪɴᴀ =1 , ꜱᴇᴄᴀ.ᴄᴏꜱᴀ =1, (ꜱɪɴᴀ/ᴄᴏꜱᴀ)= ᴛᴀɴᴀ & (ᴄᴏꜱᴀ/ꜱɪɴᴀ) =ᴄᴏᴛᴀ ]

⇒ ʟʜꜱ = 2 ᴛᴀɴᴀ ᴄᴏᴛʙ(1ᴄᴏꜱʙ+1ꜱɪɴᴀ) = 2ᴛᴀɴᴀ ᴄᴏᴛʙ ( ꜱᴇᴄʙ + ᴄᴏꜱᴇᴄᴀ ) = ʀʜꜱ. ʜᴇɴᴄᴇ, ᴘʀᴏᴠᴇᴅ