Verify the trigonometric identity:
(1+sin x)/ (1-sinx)=(1+sin x)/[cos x]
Hint: [cos x] means the absolute value of cos x
To verify the given trigonometric identity, we will simplify both sides of the equation and prove that they are equal. Here's how to do it:
Starting with the left side of the equation:
(1+sin x) / (1-sin x)
To simplify this expression, we will multiply the numerator and the denominator by the conjugate of the denominator, which is (1 + sin x). Doing so, we get:
(1+sin x) / (1-sin x) * (1+sin x)/(1+sin x)
= (1 + sin x)(1 + sin x) / (1 - sin x)(1 + sin x)
= (1 + 2sin x + sin^2x) / (1 - sin^2x)
= (1 + 2sin x + sin^2x) / cos^2x
Now, we can rewrite sin^2x as (1 - cos^2x) using the Pythagorean identity: sin^2x + cos^2x = 1. Making this substitution gives us:
= (1 + 2sin x + (1 - cos^2x)) / cos^2x
= (2 + 2sin x - cos^2x) / cos^2x
Next, let's work on simplifying the right side of the equation:
(1 + sin x) / [cos x]
We can combine the numerator using the distributive property:
= (1 + sin x) / cos x
Now, to make the denominators in both expressions the same, we can multiply the numerator and denominator of the right side by (1 - sin x):
= (1 + sin x)(1 - sin x) / cos x(1 - sin x)
= (1 - sin^2x) / (cos x - sin x*cos x)
= (1 - cos^2x) / (cos x - cos^2x)
= (2 + 2sin x - cos^2x) / cos^2x
Comparing both sides of the equation, we can see that they are equal:
(2 + 2sin x - cos^2x) / cos^2x = (2 + 2sin x - cos^2x) / cos^2x
Therefore, the trigonometric identity (1+sin x)/(1-sin x) = (1 + sin x)/cos x is verified.