Find $\sin^2 82 frac{1}{2}^\circ - \sin^2 22 frac{1}{2}^\circ$

Question

Find $\sin^2 82	frac{1}{2}^\circ - \sin^2 22	frac{1}{2}^\circ$

Accepted Answer

>

> [!TLDR]
> Use the identity $\sin^2 A - \sin^2 B = \sin(A+B)\sin(A-B)$ to collapse the difference of squares into a simple product of two known sine values.
>
> **Given:** Find $\sin^2 82	frac{1}{2}^\circ - \sin^2 22	frac{1}{2}^\circ$.
>
> **Step 1.** Use the identity $\sin^2 A - \sin^2 B = \sin(A+B)\sin(A-B)$, with $A = 82.5^\circ$, $B = 22.5^\circ$:
> $$A+B = 105^\circ, \qquad A - B = 60^\circ$$
> $$\sin^2 82.5^\circ - \sin^2 22.5^\circ = \sin 105^\circ \sin 60^\circ$$
>
> **Step 2.** Compute $\sin 105^\circ = \sin(60^\circ+45^\circ) = \sin60^\circ\cos45^\circ + \cos60^\circ\sin45^\circ$: …

Identity	Formula
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$
Quotient	$\tan \theta = \frac{\sin \theta}{\cos \theta}$ , $\cot \theta = \frac{\cos \theta}{\sin \theta}$
Reciprocal	$\csc \theta = \frac{1}{\sin \theta}$ , $\sec \theta = \frac{1}{\cos \theta}$ , $\cot \theta = \frac{1}{\tan \theta}$
Even-Odd	$\sin(-\theta) = -\sin \theta$ , $\cos(-\theta) = \cos \theta$

Q.Find $\sin^2 82\tfrac{1}{2}^\circ - \sin^2 22\tfrac{1}{2}^\circ$

Concept understanding — Trigonometric Identity Proof

Trigonometric Identity Proof: From Intuition to Precision

The Core Idea

Proving an Identity: The Method

A Simple Example

The Toolbox of Known Identities

A Slightly Harder Proof

Q.Find sin⁡28212∘−sin⁡22212∘\sin^2 82\tfrac{1}{2}^\circ - \sin^2 22\tfrac{1}{2}^\circsin28221​∘−sin22221​∘

🎓Unlock everything free for 14 days

Q.Find $\sin^2 82\tfrac{1}{2}^\circ - \sin^2 22\tfrac{1}{2}^\circ$

Unlock everything free for 14 days