Verify the following identity: sin(x)*cos(x)*(cot(x)+tan(x))=1
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cotx = cos x/sin x
tan x = sin x/ cos x
cot x + tan x = cos x/sin x + sin x/ cos x
so
cot x + sin x = cos^2 x/sin x cos x + sin^2 x/sin x cos x
cot x + tan x = (cos^2 x+sin^2x)/sin x cos x
but cos^2 x + sin^2 x = 1
you take it from there.
To verify the given trigonometric identity, we need to simplify the left-hand side (LHS) until it matches the right-hand side (RHS). Let's start by manipulating the LHS step by step:
LHS: sin(x) * cos(x) * (cot(x) + tan(x))
Recall that cot(x) is the reciprocal of tan(x). So, we can rewrite cot(x) as 1/tan(x).
LHS: sin(x) * cos(x) * (1/tan(x) + tan(x))
Now, we can find a common denominator by multiplying the two terms inside the parentheses by tan(x).
LHS: sin(x) * cos(x) * [1/tan(x) * tan(x) + tan(x) * tan(x)]
The first term simplifies to 1, and the second term simplifies to tan^2(x).
LHS: sin(x) * cos(x) * (1 + tan^2(x))
Next, we can use the trigonometric identity for sin^2(x) + cos^2(x) = 1, which can be rearranged as sin^2(x) = 1 - cos^2(x).
LHS: sin(x) * cos(x) * (1 + tan^2(x))
sin(x) * cos(x) * (sin^2(x) / cos^2(x) + tan^2(x))
Now, let's multiply the fractions by the common denominator, cos^2(x).
LHS: sin(x) * cos(x) * [(sin^2(x) + tan^2(x) * cos^2(x)) / cos^2(x)]
Using the pythagorean trigonometric identity tan^2(x) + 1 = sec^2(x), we can substitute tan^2(x) with sec^2(x) - 1.
LHS: sin(x) * cos(x) * [(sin^2(x) + (sec^2(x) - 1) * cos^2(x)) / cos^2(x)]
Expanding the numerator:
LHS: sin(x) * cos(x) * [(sin^2(x) + sec^2(x) * cos^2(x) - cos^2(x)) / cos^2(x)]
Simplifying the numerator:
LHS: sin(x) * cos(x) * [(sin^2(x) + sec^2(x) * cos^2(x) - cos^2(x)) / cos^2(x)]
sin(x) * cos(x) * [(sin^2(x) + sec^2(x) * cos^2(x) - cos^2(x)) / cos^2(x)]
sin(x) * cos(x) * [(sin^2(x) + (1/cos^2(x)) * cos^2(x) - cos^2(x)) / cos^2(x)]
sin(x) * cos(x) * [(sin^2(x) + 1 - cos^2(x)) / cos^2(x)]
sin(x) * cos(x) * (1 / cos^2(x))
(sin(x) * cos(x)) / cos^2(x)
tan(x)
Thus, the LHS simplifies to tan(x), which is equal to the RHS (1).
Therefore, the given identity sin(x) * cos(x) * (cot(x) + tan(x)) = 1 is verified.