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)

Please learn the difference between geography and geometry.