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CotA=1/tanA

tanA=1/cotA

tanA/1-cotA+cotA/1-tanA=1+secAcosecA

tanA/1-1/tanA + cotA/1-1/cotA

tanA/tanA-1/tanA + cotA/cotA-1/cotA

tanA²/tanA-1+1/tanA/1-1/tanA

tanA²/tanA-1 - 1/tanA(tanA-1)

taking lcm on both sides

tanA³-1/tanA(tanA-1)

using identity a³+b³ =(a-b)(a²+ab+b²)

(tanA-1)(tan²A+tanA+1)/tanA(tanA-1)

(tan²A +tanA+ 1)/tanA

tan²A/tanA+tanA/tanA+1/tanA

tanA+1+cot A

tanA=sinA/cos A and cosA/sinA=cotA

sinA/cosA+1+cosA/sinA

(sin²A+cos²A+cosAsinA)/cosAsinA

sin²A+cos²A=1

1/cosAsinA + 1

1/cosA=secA and 1/sinA=cosecA

secAcosecA+1

hence lhs = rhs so proves

tanA=1/cotA

tanA/1-cotA+cotA/1-tanA=1+secAcosecA

tanA/1-1/tanA + cotA/1-1/cotA

tanA/tanA-1/tanA + cotA/cotA-1/cotA

tanA²/tanA-1+1/tanA/1-1/tanA

tanA²/tanA-1 - 1/tanA(tanA-1)

taking lcm on both sides

tanA³-1/tanA(tanA-1)

using identity a³+b³ =(a-b)(a²+ab+b²)

(tanA-1)(tan²A+tanA+1)/tanA(tanA-1)

(tan²A +tanA+ 1)/tanA

tan²A/tanA+tanA/tanA+1/tanA

tanA+1+cot A

tanA=sinA/cos A and cosA/sinA=cotA

sinA/cosA+1+cosA/sinA

(sin²A+cos²A+cosAsinA)/cosAsinA

sin²A+cos²A=1

1/cosAsinA + 1

1/cosA=secA and 1/sinA=cosecA

secAcosecA+1

hence lhs = rhs so proves