How to verify the identity: (1-2cos^2(x))/(sin(x)cos(x))?
How does one verify (1-2cos^2x)/(sinxcosx) = tanx-cotx1−2cos2xsinxcosx=tanx−cotx using basic trig identities such as the Quotient Identities, the Reciprocal Identites, and the Pythagorean Identities?
How does one verify
3 Answers
Explanation:
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:
But we only want to replace one of the
And then complete the substitution:
Next, distribute the negative and simplify:
Now we can split this fraction into two:
Simplify:
Apply the quotient identities:
And you reach:
Explanation:
"using the "color(blue)"trigonometric identities"using the trigonometric identities
•color(white)(x)tanx=sinx/cosx" and "cotx=cosx/sinx∙xtanx=sinxcosx and cotx=cosxsinx
.
"consider the right side"consider the right side
tanx-cotxtanx−cotx
=sinx/cosx-cosx/sinx=sinxcosx−cosxsinx
=(sin^2x-cos^2x)/(sinxcosx)=sin2x−cos2xsinxcosx
=(1-cos^2x-cos^2x)/(sinxcosx)=1−cos2x−cos2xsinxcosx
=(1-2cos^2x)/(sinxcosx)=" left side "rArr" verified"=1−2cos2xsinxcosx= left side ⇒ verified