Pre-Calculus Prep: Trig Identity Proof -- csc^2xcos^2x = csc^2x - 1

To prove the trigonometric identity csc^2(x) * cos^2(x) = csc^2(x) - 1, we'll start with the left side of the equation and manipulate it step by step until we reach the right side.

Starting with the left side:

csc^2(x) * cos^2(x)

We can rewrite csc^2(x) as 1/sin^2(x) and cos^2(x) as (1 - sin^2(x)). Substituting these values in, we get:

(1/sin^2(x)) * (1 - sin^2(x))

Now, we'll simplify the expression by multiplying the numerators and denominators:

(1 * (1 - sin^2(x))) / (sin^2(x))

This simplifies to:

(1 - sin^2(x)) / sin^2(x)

Next, we can simplify further by using the identity 1 - sin^2(x) = cos^2(x):

cos^2(x) / sin^2(x)

Now, let's rewrite cos^2(x) as 1 - sin^2(x):

(1 - sin^2(x)) / sin^2(x)

We can split the fraction into two separate fractions:

1/sin^2(x) - sin^2(x)/sin^2(x)

Simplifying further, we get:

csc^2(x) - 1

And there you have it! We've proven that csc^2(x) * cos^2(x) is equal to csc^2(x) - 1 by manipulating the left side of the equation step by step until it matches the right side.
These videos are designed to review and reteach Precalculus and Collegeboard Pre-CALC AP content. My videos cover functions, polynomials, exponential and logarithmic expressions, trigonometry, parametric equations, polar coordinates, vectors, matrices and systems, conic sections, discrete mathematics, sequences and series; and an introduction to calculus.

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa .
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