Question

What is the value of $(1-cotx)(1+cotx)-csc^2x$?

Answer 1

Hint: We need to first simplify the expression using some certain formulae. After that we just do the simple math and use some more trigonometric formulae to solve it further. Here, we first use the formula $a^2-b^2=(a+b)(a-b)$ and after that we use a simple trigonometric identity that involves cosec and cot to further obtain the resultant expression. A rearrangement of terms is all that is required in such questions.

Complete step by step solution:
We have $(1-cotx)(1+cotx)$, we first use the formula written below to simplify this expression:
$a^2-b^2=(a+b)(a-b)$
Using this equation we obtain:
$(1-cotx)(1+cotx)=1^2-cot^2x$
$\implies (1-cotx)(1+cotx)=1-cot^2x$
Now, we need to subtract the term $csc^2x$ from this expression obtained according to the question, so we have the following situation till now:
$(1-cotx)(1+cotx)-csc^2x=1-cot^2x-csc^2x$
Now, we have the following formula:
$csc^2x=1+cot^2x$
Putting this in the equation obtained above, we get:
$(1-cotx)(1+cotx)-csc^2x=1-cot^2x-(1+cot^2x)$
Opening the brackets now in the above equation:
$\implies (1-cotx)(1+cotx)-csc^2x=1-cot^2x-1-cot^2x$
Cancelling out 1 from both sides,
$(1-cotx)(1+cotx)-csc^2x=-cot^2x-cot^2x$
Now, further adding these two negative $cot^2x$, we get the following:
$(1-cotx)(1+cotx)-csc^2x=-2cot^2x$
Hence, the solution is obtained.

Note: While opening the brackets, make sure to add a negative sign to both the terms of the expression inside the bracket. It is very common to attach the negative sign to the first term only while not changing the sign of the other quantity in the bracket as well. While using the trigonometric identities, make sure you put the sign in the right place, e.g. you might end up writing $1-cot^2x=csc^2x$, which is a wrong identity. While taking terms on the other side of the equation, make sure that you change the sign without forgetting.