sin(90-x)cos(x-180)+tanxcos(-x)sin(180+x)

Question

I need to simplify the expression without using a calculator. 

Answer 1

You need to use a couple of basic trig identities:

• sin(90-x) = cos(x)
• cos(x-180) = -cos(x)
• cos(-x) = cos(x)
• sin(180+x) = -sin(x)
• tan(x) = sin(x)/cos(x)
• cos²(x) + sin²(x) = 1

 

When you substitute these identities into your equation, you get:

 

cos(x)(-cos(x))  + (sin(x)/cos(x))(cos(x))(-sin(x))

 

Can you finish it now?

 

 

Answer 2

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

sin(x - y) = sin(x)cos(y) - cos(x)sin(y)

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

 

 

Notice that some factors have addition and subtraction of angle identities.  These factors are

 

sin(90 - x)    ,    cos(x - 180)     ,     cos(-x)  =  cos(0 - x)   ,  and     sin(180 + x)

 

 

Let's deal with these factors first.  Apply the rules for addition and subtraction identities listed above.  Then put the new forms into the expression.

 
sin(90 - x) = sin(90)cos(x) - cos(90)sin(x)

 

                 = cos(x)

 

 

cos(x - 180) = cos(x)cos(180) + sin(x)sin(180)

 

                   = -cos(x)

 

 

cos(-x) = cos(0 - x) = cos(0)cos(x) + sin(0)sin(x)

                  

                             = cos(x)

 

 

sin(180 + x) = sin(180)cos(x) + cos(180)sin(x)

 

                  = -sin(x)

 

 

Now, we can put the simplified forms into the expression.

 

sin(90 - x)cos(x - 180) + tanxcos(-x)sin(180 + x) =

 

 

cos(x)(-cos(x)) + tan(x)cos(x)(-sin(x)) =

 

-cos²(x) - tan(x)cos(x)sin(x) =

 

-cos²(x) - [(sin(x) / (cos(x)) * (cos(x)sin(x))] =

 

-cos²(x) - sin²(x) =

 

-(cos²(x) + sin²(x)) =

 

-1