Trignometric identities

Question

Simplify the expression to a single primary trig ratio?

 

(cos2x-1)/ (sin2x)

 

The answer was -tanx but I don't know why.

Please help, thank you.

Answer 1

(cos2x-1)/(sin2x)

cos2x=1-2sin²x (trig identity)

cos2x-1=1-2sin²x-1

sin2x=2sinxcosx  (trig identity)

substitute

(1-2sin²x-1)/2sinxcosx

-2sin²x/2sinxcosx

-sin²x/sinxcosx

-sinx/cosx

tan=sin/cos  (trig identity)

therefore -sinx/cosx=-tanx

Answer 2

 

Hi Harleen,

 

To get to -tanx, we need to identify which identities are equivalent to cos2x and sin2x. What identity can be substituted for cos2x and has sine in it? What identity can be substituted for sin2x and has cosine in it?

 

These substitutions will simplify the equation and give -tanx. 

Answer 3

Try starting by rewriting cos(2x) as cos²(x) - sin²(x) and sin(2x) as 2sin(x)cos(x).

 

These are your double angle formulas for sine and cosine.

 

Once you do this, you can eliminate a term in the numerator by rewriting 1 as sin²(x) + cos²(x).

 

There're likely many ways, but this will work.

 

You always want to look for a way to use the trig identities you know to your advantage when rewriting these.

Answer 4

(cos(2x)-1)/(sin(2x)) = [1-2sin²x-1]/[2sinxcosx]

 

                             = (-2sin²x)/(2sinxcosx) = -sinx/cosx = -tanx