Sinx/Cscx -1+sinx/Cscx +1= 2tan^2x
To simplify the given expression, we can start by expressing all trigonometric functions in terms of sine and cosine:
Sinx/Cscx - 1 + sinx/Cscx + 1
Using the reciprocal identities, we can rewrite the expression as:
sinx/sinx·(1/sinx) - 1 + sinx/sinx·(1/sinx) + 1
Now, simplifying each term, we get:
1 - 1 + 1 + 1
This simplifies to:
2
On the other side of the equation, we have:
2tan^2x
Using the identity tan^2x = sin^2x / cos^2x, we can rewrite the expression as:
2(sin^2x / cos^2x)
Now, to simplify the expression even further, we can use the identity sin^2x + cos^2x = 1. Rearranging this identity, we get:
sin^2x = 1 - cos^2x
Substituting this into the expression, we have:
2((1 - cos^2x) / cos^2x)
Expanding the expression, we get:
2/cos^2x - 2
Using the identity 1/cos^2x = sec^2x, we can simplify further:
2sec^2x - 2
The expression on the right side of the equation is 2sec^2x, whereas the expression on the left side simplifies to 2. Since 2 is equal to 2sec^2x, we can conclude that the given equation is true.