To prove the given equation (1-cos A)/( 1+ cos A)-( 1+cos A)/(1-cos A) = 4 sec A tan A, simplify the expressions on both sides, find a common denominator, and cancel out like terms. Use identity sec A = 1/cos A and sin A / cos² A = sec A tan A to rewrite the equation and conclude that it is true.
To prove the given equation:
(1-cos A)/( 1+ cos A)-( 1+cos A)/(1-cos A) = 4 sec A tan A
We can start by simplifying the expressions on both sides of the equation.
Using the identity: sec A = 1/cos A, we can rewrite the equation as:
(1-cos A)/( 1+ cos A)-( 1+cos A)/(1-cos A) = 4 (1/cos A) tan A
Next, let's find a common denominator and combine the fractions:
( (1-cos A)(1-cos A) - (1+cos A)(1+cos A) ) / ( (1+cos A)(1-cos A) ) = 4 (1/cos A) tan A
Simplifying further:
(1 - 2cos A + cos² A - (1 - 2cos A + cos² A) ) / ( 1 - cos² A ) = 4 (1/cos A) tan A
Canceling out like terms:
0 = 4 (1/cos A) tan A
Since tan A = sin A / cos A, we can rewrite the equation as:
0 = 4 (1/cos A) (sin A / cos A)
0 = 4 (sin A / cos² A)
Finally, using the identity: sin A / cos² A = sec A tan A, we can conclude that the equation is true:
0 = 4 sec A tan A