TRIGONOMETRY PROBLEM WITH SOLUTION

1.Prove the indentity  tan²(x) - sin²(x) = tan²(x) sin²(x)

1. Use the identity tan(x) = sin(x) / cos(x) in the left hand side of the given identity. 
   
   tan²(x) - sin²(x) = sin²(x) / cos²(x) - sin²(x) 
   
   = [ sin²(x) - cos²(x) sin²(x) ] / cos²(x) 
   
   = sin²(x) [ 1 - cos²(x) ] / cos²(x) 
   
   = sin²(x) sin²(x) / cos²(x) 
   
   = sin²(x) tan²(x) which is equal to the right hand side of the given identity.

2.Prove the indentity 

(1 + cos(x) + cos(2x)) / (sin(x) + sin(2x)) = cot(x)

1. Use the identities cos(2x) = 2 cos²(x) - 1 and sin(2x) = 2 sin(x) cos(x) in the left hand side of the given identity. 
   
   [ 1 + cos(x) + cos(2x) ] / [ sin(x) + sin(2x) ] 
   
   = [ 1 + cos(x) + 2 cos²(x) - 1 ] / [ sin(x) + 2 sin(x) cos(x) ] 
   
   = [ cos(x) + 2 cos²(x) ] / [ sin(x) + 2 sin(x) cos(x) ] 
   
   = cos(x) [1 + 2 cos(x)] / [ sin(x)( 1 + 2 cos(x) ) ] 
   
   = cot(x)