Question

AHOLA BRAINLICS!!!!!!----------------❤️❤️❤️❤️ PROVE THAT SinA( 1 + TanA) + cosA(1 + cotA) = SecA+ CosecA-------- NO SPAMMING❎❎❎❎ CONTENT QUALITY NEEDED‼️‼️

Answer 1

sin A + sin^2A/cosA + cosA + cos^2/Sin a

sin^3 +cos^3)/sinA Cosa + Sin a + cosa

( sin+ cos)^3 - 3sincos( sin + cos) ) /sin cos +sin +cos

sin+ cos)^3/sincos - 3( sin + cos) + sin + cos

sin + cos( (sin+cos)^2/sin cos - 3 + 1)

sin + cos( (sin^2 + cos^2 + 2sincos)/sin cos -2)

sin+ cos( cosec sec + 2 -2)

sin + cos ( 1/sin cos)

cosec + sec

Answer 2

Here is your solution

LHS.

$SinA( 1 + TanA) + cosA(1 + cotA) \\\\ =sin A + sin^2A/cosA + cosA +\\ cos^2A/SinA \\ \\ =(sin^3A +cos^3A)/sinA CosA + SinA + cosA \\ \\ =( sinA+ cosA)^3 - 3sinAcosA( sinA + cosA) ) /sinAcosA \\+sinA +cosA \\ \\ =(sinA+ cosA)^3/sinAcosA - 3( sinA + cosA) \\+ sinA + cosA \\ \\ =sinA + cosA(sinA+cosA)^2/sinA\\ cosA - 3 + 1) \\ \\ =sinA+ cosA(sin^2A + cos^2A +\\ 2sinAcosA)/sinAcosA-2) \\ \\ =sinA + cosA( cosecAsecA + 2 -2) \\ \\ =sinA + cosA ( 1/sinAcosA) \\ \\ cosecA + secA\: proved$

Note:-

Always try to break tan in sin /cos to simply easily.

Hope it helps you