Verify that tan(2x) = 2sinxcosxsec(2x)

Saturday, December 12, 2015

Verify that tan(2x) = 2sinxcosxsec(2x)

$\displaystyle{\tan{{\left({2}{x}\right)}}}={2}{\sin{{x}}}{\cos{{x}}}{\sec{{2}}}{x}$

To verify this identity, simplify the right side of the equation until it matches the left side.

(1) This verification will include the Double Angle formula:

$\displaystyle{\sin{{\left({2}{x}\right)}}}={2}{\sin{{x}}}{\cos{{x}}}$

(2) This verification will include the Reciprocal Identity:

$\displaystyle{\sec{{\left({x}\right)}}}=\frac{{1}}{{\cos{{\left({x}\right)}}}}$

(3) The verification will include the Quotient Identity:

$\displaystyle{\tan{{\left({x}\right)}}}=\frac{{\sin{{\left({x}\right)}}}}{{\cos{{\left({x}\right)}}}}$

Here are the steps.

$\displaystyle{\tan{{\left({2}{x}\right)}}}={2}{\sin{{\left({x}\right)}}}{\cos{{\left({x}\right)}}}{\sec{{\left({2}{x}\right)}}}$

$\displaystyle{\tan{{\left({2}{x}\right)}}}={\sin{{\left({2}{x}\right)}}}{\sec{{\left({2}{x}\right)}}}$ [Using the Double Angle Formula]

$\displaystyle{\tan{{\left({2}{x}\right)}}}={\sin{{\left({2}{x}\right)}}}\cdot\frac{{1}}{{\cos{{\left({2}{x}\right)}}}}$ [Using the Reciprocal Identity]

$\displaystyle{\tan{{\left({2}{x}\right)}}}=\frac{{\sin{{\left({2}{x}\right)}}}}{{\cos{{\left({2}{x}\right)}}}}$

$\displaystyle{\tan{{\left({2}{x}\right)}}}={\tan{{\left({2}{x}\right)}}}$ ...