Prove the following identities

Question

A)cotA+cosA=(cosA(1+sinA))/sinA

B)sec^2A-sin^2A=cos^2A+tan^2A

C)cscA(1-cosA)=(sinA+sinAcosA)/cos^2A+2cosA+1

Answer 1

A)

 

Work out the left side.

 

(cosA / sinA) + cosA =

 

(cosA + sinAcosA) / sinA =

 

cosA(1 + sinA) / sinA

 

 

B)

 

Work the left side.

 

sec²A - sin²A =

 

(1 / cos²A) - sin²A =

 

(1 - sin²Acos²A) / cos²A =

 

(sin²A + cos²A - sin²Acos²A) / cos²A =

 

tan²A + 1 - sin²A =

 

tan²A + cos²A

 

 

C)

 

Work out the right side, since it has more to work with.

 

(sinA + sinAcosA) / (cos²A + 2cosA + 1)

 

 

You can factor out both numerator and denominator.

 

sinA(1 + cosA) / (cosA + 1)²

 

(cosA + 1) cancel each other out.  You are left with

 

 

sinA / (1 + cosA)

 

 

Multiply numerator and denominator by conjugate of denominator to get

 

sinA(1 - cosA) / (1 - cos²A) =

 

sinA(1 - cosA) / sin²A =

 

(1 - cosA) / sinA =

 

cscA(1 - cosA)