3)Show that (cot^2)x - (cos^2)x = (cos^2)x(cot^2)x
L.S.
= cos^2 x/sin^2 x - cos^2 x
= (cos^2 x - (sin^2 x)(cos^2 x))/sin^2 x
= cos^2 x(1 - sin^2 x)/sin^2 x
= cos^2 x(cos^2 x)/sin^2 x
= (cos^2 x)(cot^2 x)
= R.S.
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To prove that (cot^2)x - (cos^2)x = (cos^2)x(cot^2)x, we can use the trigonometric identities for cotangent and cosine.
Let's start with the left side of the equation:
(cot^2)x - (cos^2)x
Using the identity cot^2(x) = 1/(tan^2(x)) and tan^2(x) = 1 - cos^2(x), we can rewrite the equation as:
1/(tan^2(x)) - (cos^2(x))
Next, using the identity sin^2(x) + cos^2(x) = 1, we can express tan^2(x) in terms of sin(x) and cos(x):
1/(1 - cos^2(x)) - (cos^2(x))
To combine the fractions, we need a common denominator. Multiplying the first fraction by (1 - cos^2(x))/(1 - cos^2(x)), we get:
(1 - cos^2(x))/(1 - cos^2(x)) - (cos^2(x))
Now, simplifying the numerator, we have:
1 - cos^2(x) - (cos^2(x))
Combining like terms, we get:
1 - 2cos^2(x)
Now, let's simplify the right side of the equation:
(cos^2(x))(cot^2(x))
Using the identity cot^2(x) = 1/(tan^2(x)), we can rewrite it as:
(cos^2(x))(1/(tan^2(x)))
Substituting tan^2(x) = 1 - cos^2(x) again, we get:
(cos^2(x))(1/(1 - cos^2(x)))
Multiplying the fractions, we obtain:
(cos^2(x))/(1 - cos^2(x))
Using the identity sin^2(x) + cos^2(x) = 1, we can simplify the denominator:
(cos^2(x))/(sin^2(x))
Again, using the identity cos^2(x) = 1 - sin^2(x), we rewrite the equation as:
(1 - sin^2(x))/(sin^2(x))
Combining like terms, we get:
1/sin^2(x) - 1
Finally, using the identity csc^2(x) = 1/sin^2(x), we express it as:
csc^2(x) - 1
Now, we can see that the right side simplifies to the same expression as the left side:
(cot^2)x - (cos^2)x = (cos^2)x(cot^2)x.
Hence, we have shown that (cot^2)x - (cos^2)x = (cos^2)x(cot^2)x through the step-by-step derivation of both sides of the equation.