Final answer:
To prove the trigonometric identity tanA - CotA = -2Cot2A, we manipulate the left side using trigonometric identities to match the form of the right side, eventually showing the two sides are equivalent.
Step-by-step explanation:
To prove the trigonometric identity tanA - CotA = -2Cot2A, we will transform the left side of the equation to look like the right side by using trigonometric identities and relationships.
Firstly, remember that tanA = sinA/cosA and CotA = cosA/sinA. Now let's rewrite the left-hand side of the equation:
tanA - CotA = (sinA/cosA) - (cosA/sinA)
To combine these fractions, we need a common denominator:
(sinA/cosA) - (cosA/sinA) = (sin2A - cos2A) / (sinA*cosA)
Using the Pythagorean identity which is sin2A + cos2A = 1, we can rewrite sin2A - cos2A as -(cos2A - sin2A).
So we have: -(cos2A - sin2A) / (sinA*cosA)
Now, we use the double angle identities: cos2A = cos2A - sin2A and tan2A = 2tanA / (1 - tan2A). This allows us to rewrite cos2A as 1 - 2sin2A or 2cos2A - 1.
Since Cot2A is the reciprocal of tan2A, we can write:
Cot2A = (1 - tan2A)/(2tanA)
But tanA is sinA/cosA, so:
Cot2A = (cos2A - sin2A)/(2sinA*cosA)
Finally, since Cot2A = (cos2A - sin2A)/(2sinA*cosA), we see that the original expression simplifies to:
(sin2A - cos2A) / (sinA*cosA) = -2 * (cos2A - sin2A) / (2sinA*cosA)
Which simplifies to:
tanA - CotA = -2Cot2A
Hence, we have proven the identity.