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Simplify the following expression:

[tex]\[ \frac{1 - \cos y}{\sin y} + \frac{\sin y}{1 - \cos y} = 2 \csc y \][/tex]


Sagot :

To prove the given trigonometric identity:

[tex]\[ \frac{1 - \cos y}{\sin y} + \frac{\sin y}{1 - \cos y} = 2 \csc y \][/tex]

We'll start by simplifying the left-hand side (LHS) of the equation and show that it equals the right-hand side (RHS).

### Step-by-Step Solution:

1. Write down the left-hand side (LHS):

[tex]\[ \text{LHS} = \frac{1 - \cos y}{\sin y} + \frac{\sin y}{1 - \cos y} \][/tex]

2. Find a common denominator for the fractions:

To simplify the LHS, we get a common denominator, which would be [tex]\( \sin y (1 - \cos y) \)[/tex]:

[tex]\[ \text{LHS} = \frac{(1 - \cos y)^2 + \sin^2 y}{\sin y (1 - \cos y)} \][/tex]

3. Simplify the numerator:

Notice that we can rewrite [tex]\( (1 - \cos y)^2 \)[/tex] as [tex]\( 1 - 2\cos y + \cos^2 y \)[/tex]:

[tex]\[ (1 - \cos y)^2 = 1 - 2 \cos y + \cos^2 y \][/tex]

Also, recall the Pythagorean identity:

[tex]\[ \sin^2 y + \cos^2 y = 1 \][/tex]

Using this identity, substitute [tex]\( \cos^2 y \)[/tex]:

[tex]\[ \sin^2 y + \cos^2 y = 1 \quad \Rightarrow \quad 1 - \sin^2 y = \cos^2 y \][/tex]

Substituting these identities back into the numerator:

[tex]\[ (1 - \cos y)^2 + \sin^2 y = (1 - 2 \cos y + \cos^2 y) + \sin^2 y = 1 - 2 \cos y + \cos^2 y + \sin^2 y = 1 - 2 \cos y + 1 = 2 - 2 \cos y \][/tex]

4. Factor out the common term in the numerator:

[tex]\[ \text{Above Proposition} = 2 (1 - \cos y) \][/tex]

Thus, the numerator simplifies to:

[tex]\[ \frac{2 (1 - \cos y)}{\sin y (1 - \cos y)} \][/tex]

5. Cancel the common term [tex]\( (1 - \cos y) \)[/tex] in the numerator and the denominator:

[tex]\[ \frac{2 (1 - \cos y)}{\sin y (1 - \cos y)} = \frac{2}{\sin y} = 2 \csc y \][/tex]

We see that the left-hand side (LHS) simplifies to:

[tex]\[ \text{LHS} = 2 \csc y \][/tex]

6. Compare it with the right-hand side (RHS):

[tex]\[ \text{RHS} = 2 \csc y \][/tex]

7. Conclusion:

Since the simplified LHS is equal to the RHS, we have proven the given identity:

[tex]\[ \frac{1 - \cos y}{\sin y} + \frac{\sin y}{1 - \cos y} = 2 \csc y \][/tex]