cos2x/sinx+sin2x/cosx=cscx

Question

for all values of x for which the expressions are defined, prove the following is an identity. thank you.

Answer 1

The key idea here is the cosine sum formula: cos(x+y)=cos(x)cos(y)-sin(x)sin(y). We are going to apply it to cos(x)=cos(2x-x)=cos(2x+(-x)). After we apply the formula, we get

 

cos(x) = cos(2x)cos(-x)-sin(2x)sin(-x).

 

Since cos is an even function, cos(-x)=cos(x)

Since sin is an odd function, sin(-x)=-sin(x)

 

This simplifies to

 

cos(x) = cos(2x)cos(x)+sin(2x)sin(x).

 

Now, we need to divide both sides of the equation by sin(x)cos(x). (Note that the original identity can only be valid when both sin(x) and cos(x) are not zero, so we can safely assume that we are not dividing by zero here.)

 

On the left hand side, we get

 

cos(x)/(cos(x)*sin(x)).

 

The cosines cancel, and we are left with 1/sin(x)=csc(x)

 

On the right and side, we have cos(2x)cos(x)/(sin(x)cos(x))+sin(2x)sin(x)/(sin(x)cos(x))=cos(2x)/sin(x)+sin(2x)/cos(x).

 

We have shown that this is equal to csc(x), which is what was required. I hope this helps.