How to verify the identity: (1-2cos^2(x))/(sin(x)cos(x))?

How does one verify 12cos2xsinxcosx=tanxcotx using basic trig identities such as the Quotient Identities, the Reciprocal Identites, and the Pythagorean Identities?

3 Answers
Feb 25, 2018

tanxcotx

Explanation:

12cos2(x)sin(x)cos(x)=

(sin2(x)+cos2(x))2cos2(x)sin(x)cos(x)=

sin2(x)cos2(x)sin(x)cos(x)=

sin2xsinxcosxcos2xsinxcosx=

sinxcosxcosxsinx=

tanxcotx

Feb 25, 2018

Please look at the Explanation area for this is a "how" question.

Explanation:

The first step to this problem is to use a Pythagorean Identity:

cos2x=1sin2x

But we only want to replace one of the cos2x so we can rewrite the identity like this for clarity:

1(cos2x+cos2x)sinxcosx=tanxcotx

And then complete the substitution:

1(1sin2x+cos2x)sinxcosx=tanxcotx

Next, distribute the negative and simplify:

sin2xcos2xsinxcosx=tanxcotx

Now we can split this fraction into two:

sin2xsinxcosxcos2xsinxcosx=tanxcotx

Simplify:

sinxcosxcosxsinx=tanxcotx

Apply the quotient identities:

sinxcosx=tanx and cosxsinx=cotx

And you reach:

tanxcotx=tanxcotx

Feb 25, 2018

see explanation

Explanation:

using the trigonometric identities

xtanx=sinxcosx and cotx=cosxsinx

.xsin2x+cos2x=1

consider the right side

tanxcotx

=sinxcosxcosxsinx

=sin2xcos2xsinxcosx

=1cos2xcos2xsinxcosx

=12cos2xsinxcosx= left side verified