sin 2x(cot x + tan x) = 2
(verify the identity)
LS
= sin 2x(cot x + tan x) = 2
= 2sinxcosx(cosx/sinx + sinx/cosx)
= 2cos^2x + 2sin^2x
= 2(cos^2x + sin^2x)
= 2(1) = 2
= RS
To verify the identity sin 2x(cot x + tan x) = 2, we need to simplify both sides of the equation to check if they are equal.
Starting with the left side of the equation:
1. Recall the double-angle identity for sine: sin 2x = 2sin xcos x.
2. Substitute sin 2x with 2sin xcos x in the equation: 2sin xcos x(cot x + tan x).
Now, let's simplify the left side of the equation further:
3. Expand the expression: 2sin xcos x(cot x + tan x) = 2sin xcos x * cot x + 2sin xcos x * tan x.
4. Apply the identity cot x = cos x / sin x and tan x = sin x / cos x: 2sin xcos x * cot x + 2sin xcos x * tan x = 2sin xcos x * (cos x / sin x) + 2sin xcos x * (sin x / cos x).
5. Simplify: 2sin xcos x * (cos x / sin x) + 2sin xcos x * (sin x / cos x) = 2cos x + 2sin x.
Now, let's simplify the right side of the equation:
6. The right side of the equation is already simplified as 2.
Comparing the left and right sides of the equation:
7. We have 2cos x + 2sin x = 2.
Since the left side of the equation simplifies to 2 and it matches the right side of the equation, we can conclude that sin 2x(cot x + tan x) = 2 is indeed an identity.
Note: In this explanation, we assumed x ≠ (2n + 1)π/2, where n is an integer, to avoid division by zero when using the identity cot x = cos x / sin x and tan x = sin x / cos x.