Verify the following identity:
(sin2x)/sin - (cos2x)/cosx=secx
LS = (sin2x)/sin - (cos2x)/cosx
= 2sinxcox/sinx - (2cos^2 x - 1)/cosx
= 2cosx - (2cos^2 x/cosx - 1/cosx)
= 2cosx - 2cosx + 1/cosx
= secx
= RS
To verify the given identity:
1. Start by expressing sin2x and cos2x in terms of sine and cosine functions.
sin2x = 2sinxcosx
cos2x = cos^2(x) - sin^2(x)
2. Substitute the expressions for sin2x and cos2x in the given identity:
(2sinxcosx)/sin - (cos^2(x) - sin^2(x))/cosx = secx
3. Simplify the terms on the left-hand side of the equation:
Multiply both terms by the least common denominator, which is sin(x)cos(x):
2cos^2(x) - (cos^2(x) - sin^2(x)) = sin(x)cos(x)secx
4. Continue simplifying:
Distribute the negative sign:
2cos^2(x) - cos^2(x) + sin^2(x) = sin(x)cos(x)secx
Combine like terms:
cos^2(x) + sin^2(x) = sin(x)cos(x)secx
5. Use the Pythagorean identity sin^2(x) + cos^2(x) = 1:
1 = sin(x)cos(x)sec(x)
6. Rearrange the terms to isolate sec(x) on one side:
sec(x) = 1/(sin(x)cos(x))
Thus, we have verified that (sin2x)/sin - (cos2x)/cosx is equal to secx by simplifying the expression step by step.