Verify the identity:
tanx(cos2x) = sin2x - tanx
Left Side = (sinx/cosx)(2cos^2 x -1)
=sinx(2cos^2 x - 1)/cosx
Right Side = 2sinx cosx - sinx/cosx
=(2sinxcos^2 x - sinx)/cosx
=sinx(2cos^2 x -1)/cosx
= L.S.
Q.E.D.
To verify the given identity, we start by simplifying both sides of the equation.
Starting with the left side:
tanx(cos2x)
Using the trigonometric identity for the double angle of cosine, cos2x = 2cos^2x - 1:
tanx(2cos^2x - 1)
Next, we simplify the expression on the right side:
sin2x - tanx
Using the trigonometric identity for the double angle of sine, sin2x = 2sinxcosx:
2sinxcosx - tanx
Now, let's further simplify the left side expression:
tanx(2cos^2x - 1)
= (sinx/cosx)(2cos^2x - 1)
= sinx(2cos^2x - 1)/cosx
And simplify the right side expression:
2sinxcosx - tanx
Now, we can see that both the left side and the right side simplify to the same expression:
sinx(2cos^2x - 1)/cosx = 2sinxcosx - tanx
Hence, we have shown that the left side is equal to the right side, and the given identity is verified.
Q.E.D.