Proving an Identity: Show LHS Equals RHS

What you do

You shape the function y = (1 - sin(x)) * (1 + sin(x)) - k * cos(x)^2 and watch the graph answer.

Challenges to clear

• LHS = (1−sin x)(1+sin x) = 1 − sin²x. Slide k so k·cos²x matches it: only k = 1 gives 0 at x = 1.
• Still 0 at x = 0.5 — LHS equals RHS everywhere: identity proved.
• Double the left-hand side: slide a to 2, giving 2(1 − sin x)(1 + sin x).
• That expands to 2(1 − sin²x) = 2cos²x. Slide k until the difference is 0 at x = 1 — the identity scales with it.

Check yourself

Why does the expression (1 - sin x)(1 + sin x) simplify to 1 - sin^2 x?

How does the Pythagorean identity sin^2 x + cos^2 x = 1 help complete the proof?

• It follows the difference of squares pattern: (a-b)(a+b) = a^2 - b^2. — correct
• Because sin x and -sin x cancel each other out directly.
• Because multiplying by 1 leaves the term unchanged.
• Because sin^2 x is always equal to 1.
• It allows us to substitute 1 - sin^2 x with cos^2 x. — correct
• It proves that sin x and cos x are always equal.
• It shows that sin^2 x + cos^2 x equals 2.
• It allows us to divide both sides by sin x.

Think about it

• (1 − sin)(1 + sin) = 1 − sin². By the Pythagorean identity, that equals how many cos²?