Answer:
Explanation:
Starting with the left-hand side (LHS) of the identity:
(cos x)(tan x + sin x cot x)
We can simplify using trigonometric identities. First, we can rewrite tan x as sin x/cos x and cot x as cos x/sin x:
(cos x)(sin x/cos x + sin x(cos x/sin x))
Simplifying, we get:
(sin x + cos2 x)/sin x
Next, we can multiply the numerator and denominator by the conjugate of sin x + cos2 x, which is sin x - cos2 x:
[(sin x + cos2 x)/(sin x + cos2 x)] * [(sin x - cos2 x)/(sin x - cos2 x)]
Expanding and simplifying, we get:
[(sin x)(sin x - cos2 x) + (cos2 x)(sin x - cos2 x)] / [(sin x)(sin x - cos2 x) + (cos2 x)(sin x + cos2 x)]
Simplifying further, we get:
(sin2 x - cos2 x + cos2 x sin x - cos4 x) / (sin2 x + cos2 x)
Using the identity sin2 x + cos2 x = 1 and simplifying, we get:
(sin x - cos2 x) / 1
Finally, using the identity cos2 x = 1 - sin2 x, we can simplify to:
(sin x - (1 - sin2 x)) / 1
Simplifying further, we get:
2 sin x - 1
So the left-hand side is equal to 2 sin x - 1.
Now, let's simplify the right-hand side (RHS) of the identity:
sin x + cos2 x
Using the identity cos2 x = 1 - sin2 x, we can rewrite the RHS as:
sin x + 1 - sin2 x
Simplifying further, we get:
1 - sin2 x + sin x
Finally, using the identity sin2 x + cos2 x = 1, we can rewrite as:
cos2 x + sin x
So the right-hand side is equal to cos2 x + sin x.
Since the left-hand side is equal to 2 sin x - 1 and the right-hand side is equal to cos2 x + sin x, the identity is not true in general. Therefore, the given identity does not hold true.
hope this helps