Simplify completely into an expression with sin (A) or cos (A) only:

Question

1 - (sin(A) + cos(A))^2

Answer 1

1 - (sinA + cosA)(sinA + cosA) =

 

1 - (sin²A + 2sinAcosA + cos²A) =

 

1 - sin²A - 2sinAcosA - cos²A

 

 

Use the trig identities:

 

sin²A + cos²A = 1

 

cosA = √(1 - sin²A)
       
 
Let's put this entire expression in terms of sinA.
 
1 - sin²A - 2sinA√(1 - sin²A) - (1 - sin²A) =
 
1 - sin²A - 2sinA√(1 - sin²A) - 1 + sin²A =
 
-2sinA * √(1 - sin²A) =
 
-2sinA * √[(1 + sinA)(1 - sinA)]

Answer 2

1-sinA+cos²A

=1-sinA+1-sin²A ..... as sin²A+cos²A=1

=2-sinA-sin²A

=2+sinA-2sinA-sin²A............. adjusting operators

=(2+sinA)-sinA(2+sinA)............. adjusting operators

=(2+sinA)(1-sinA)............. adjusting operators