cosec theta cosec theta
Cosec theta-1 + cosec theta+1
=2sec square theta
I can make no sense of your work.
Apparently this is a bizarre attempt to write a fraction.
Try using parentheses and virgules, as in
(cscθ+1)/(cscθ-1)
or something
To prove the given trigonometric identity, we'll manipulate the left-hand side of the equation using trigonometric properties and identities.
Starting with the left-hand side of the equation:
cosec(theta) / (cosec(theta - 1) + cosec(theta + 1))
To simplify this expression, we'll first find the least common denominator (LCD) for the two cosec terms in the denominator:
LCD = cosec(theta - 1) * cosec(theta + 1)
Now, let's multiply the numerator and denominator by the LCD:
cosec(theta) * (cosec(theta - 1) * cosec(theta + 1)) / (cosec(theta - 1) * cosec(theta + 1))
Using the property that cosec(theta) = 1/sin(theta), we can rewrite the numerator:
(1/sin(theta)) * (cosec(theta - 1) * cosec(theta + 1)) / (cosec(theta - 1) * cosec(theta + 1))
Next, we'll use the property that cosec(theta) * sin(theta) = 1:
(1/sin(theta)) * (cosec(theta - 1) * cosec(theta + 1)) / (sin(theta - 1) * sin(theta + 1))
Now, we'll use the property that cosec(theta) * sin(theta) = 1 again to simplify the denominator:
(1/sin(theta)) * (cosec(theta - 1) * cosec(theta + 1)) / (cosec(theta - 1) * cosec(theta + 1))
Notice that the denominator cancels out with the numerator, leaving us with:
1/sin(theta)
Recall that sec(theta) = 1/cos(theta). We can square both sides of this equation to get:
(sec(theta))^2 = (1/cos(theta))^2
(sec(theta))^2 = 1/(cos(theta))^2
sec^2(theta) = 1/cos^2(theta)
Using the reciprocal property of cos and sin (cos(theta) = 1/sec(theta) and sin(theta) = 1/cosec(theta)), we can rewrite this as:
sec^2(theta) = sin^2(theta)
Finally, since sec^2(theta) = sin^2(theta), we can substitute this in the previous result:
1/sin(theta) = sec^2(theta)
Thus, we have proved that:
cosec(theta) / (cosec(theta - 1) + cosec(theta + 1)) = 2sec^2(theta)