show that 1-sin x/cosx + cosx/1-sin x = 2 sec x
To prove that the given equation 1 - sin x / cos x + cos x / 1 - sin x = 2 sec x is true, we need to simplify both sides of the equation and show that they are equal.
Let's start by simplifying the left-hand side of the equation:
1 - sin x / cos x + cos x / 1 - sin x
To combine the fractions with different denominators, we need to find a common denominator. In this case, the common denominator is (cos x)(1 - sin x):
(1 - sin x)(1 - sin x) / (cos x)(1 - sin x) + (cos x)(cos x) / (1 - sin x)(cos x)
Simplifying further:
(1 - sin^2 x) / (cos x - sin x cos x) + cos^2 x / (cos x - sin x cos x)
Using the identity sin^2 x + cos^2 x = 1:
(1 - cos^2 x) / (cos x - sin x cos x) + cos^2 x / (cos x - sin x cos x)
Now, factoring out (cos x - sin x cos x) as a common denominator:
[(1 - cos^2 x) + cos^2 x] / (cos x - sin x cos x)
Simplifying the numerator:
(1 - cos^2 x + cos^2 x) / (cos x - sin x cos x)
The terms -cos^2 x and cos^2 x cancel out:
1 / (cos x - sin x cos x)
Now, multiply both the numerator and the denominator by sec x:
[1 * sec x] / [(cos x - sin x cos x) * sec x]
Using the identity sec x = 1 / cos x:
sec x / (cos x - sin x cos x)
Factor out cos x from the denominator:
sec x / [cos x(1 - sin x)]
Using the identity 1 - sin x = cos^2 x:
sec x / [cos x(cos^2 x)]
Now, cancel out the common factor of cos x:
sec x / (cos^3 x)
Using the identity sec x = 1 / cos x:
(1 / cos x) / (cos^3 x)
Dividing the fractions:
1 / (cos x * cos^3 x)
Using the identity cos^3 x = (cos x)^3:
1 / (cos^4 x)
Recall that sec x is defined as 1 / cos x:
1 / (cos^4 x) = 1 / (sec^4 x)
Therefore, the left-hand side of the equation simplifies to:
1 - sin x / cos x + cos x / 1 - sin x = 1 / (sec^4 x) = sec^(-4) x
Since sec^(-4) x is the same as 1 / (sec^4 x) and the right-hand side of the equation is 2 sec x, we can see that:
sec^(-4) x = 2 sec x
Taking the reciprocal of both sides of the equation:
1 / sec^4 x = 2 / sec x
Using the fact that sec x = 1 / cos x:
cos^4 x = 2 cos x
Dividing both sides of the equation by cos x:
cos^3 x = 2
Since the left-hand side of the equation is a cube, we can conclude that:
cos x = cube root of 2
Therefore, the equation is true for the values of x where cos x is equal to the cube root of 2.