Proving the Trigonometric Identity ( frac{1}{1-cos x} cdot frac{1}{1 cos x} 2csc^2 x )

Understanding and proving trigonometric identities is a crucial skill in mathematics, particularly in calculus and advanced algebra. One such identity that often appears in these contexts is:

( frac{1}{1-cos x} cdot frac{1}{1 cos x} 2csc^2 x )

Step 1: Combining the Fractions

First, let's start by combining the fractions on the left-hand side (LHS).

Consider the expression: (frac{1}{1 - cos x} cdot frac{1}{1 cos x} frac{1(1 cos x) 1 - cos x}{(1 - cos x)(1 cos x)} frac{2}{1 - cos^2 x})

Step 2: Using the Difference of Squares

The denominator can be simplified using the difference of squares formula.

Recall the difference of squares formula: (a^2 - b^2 (a - b)(a b)) Apply it to the denominator: (1 - cos^2 x (1 - cos x)(1 cos x)) Thus, we have: (frac{2}{(1 - cos x)(1 cos x)} frac{2}{1 - cos^2 x})

Step 3: Using Trigonometric Identities

Now, let's use a well-known trigonometric identity to further simplify the expression.

Recall the Pythagorean identity: [cos^2 x sin^2 x 1] Rearrange the identity: [sin^2 x 1 - cos^2 x] Substitute this into the denominator: (frac{2}{1 - cos^2 x} frac{2}{sin^2 x})

Step 4: Converting to Cosecant

Finally, let's convert the expression to use the cosecant function:

Recall that: [csc x frac{1}{sin x}] Thus: [csc^2 x left(frac{1}{sin x}right)^2 frac{1}{sin^2 x}] Therefore: [frac{2}{sin^2 x} 2 csc^2 x]

Conclusion: Left-Hand Side Right-Hand Side

Through algebraic manipulation and the use of trigonometric identities, we have successfully proven that:

(frac{1}{1 - cos x} cdot frac{1}{1 cos x} frac{2}{1 - cos^2 x} frac{2}{sin^2 x} 2 csc^2 x)

Related Keywords

trigonometric identities proving identities csc

Understanding and practicing these steps will not only help in solving similar problems but also deepen your overall comprehension of trigonometric functions and their relationships.