Final answer:
By substituting trigonometric identities into the given expression and then simplifying and rearranging the terms, we prove that cosec x + (2 cos x) / (1 + cot x) equals sin x + cos x.
Step-by-step explanation:
To prove that cosec x + (2 cos x) / (1 + cot x) equals sin x + cos x, we need to manipulate the left-hand side of the equation using trigonometric identities.
First, we express everything in terms of sine and cosine:
- cosec x = 1/sin x
- cot x = cos x/sin x
By substituting these into our expression, we get:
(1/sin x) + (2 cos x) / (1 + (cos x/sin x))
Combining the terms over a common denominator, we get:
(sin x + 2 cos2 x) / sin x
Using the Pythagorean identity sin2 x + cos2 x = 1, we can rewrite 2 cos2 x as 2 (1 - sin2 x), giving us:
(sin x + 2 (1 - sin2 x)) / sin x
Simplify the expression:
(sin x + 2 - 2 sin2 x) / sin x = (2 - sin x) / sin x
Dividing each term by sin x:
2/sin x - sin x/sin x = 2 cosec x - 1
Now, we can add 1 on both sides to get:
2 cosec x = sin x + 1
And since 2 cosec x = 2/sin x, and 2/sin x = sin x + 1, the original expression simplifies to:
sin x + cos x
Thus, proving that cosec x + (2 cos x) / (1 + cot x) = sin x + cos x.