Trigonometric Identity Verifier
Trigonometry
Intro: Rewrite with Pythagorean, angle-sum, double-angle, and product-to-sum rules.
Worked example
- Verify $\sin^2 x + \cos^2 x = 1$
- Goal: Show that the left-hand side (LHS) $\sin^2 x + \cos^2 x$ is equal to the right-hand side (RHS) $1$.
- Step 1 — Recall the definitions of sine and cosine on the unit circle: for an angle $x$, a point on the unit circle is $(\cos x, \sin x)$.
- Step 2 — By the Pythagorean Theorem, every point $(\cos x, \sin x)$ on the unit circle satisfies $\cos^2 x + \sin^2 x = 1$.
- Step 3 — This is the geometric reasoning. To verify algebraically, start with the identity $\sin x = \dfrac{\text{opposite}}{\text{hypotenuse}}$ and $\cos x = \dfrac{\text{adjacent}}{\text{hypotenuse}}$ in a right triangle.
- Step 4 — Square each: $\sin^2 x = \dfrac{\text{opposite}^2}{\text{hypotenuse}^2}$, $\cos^2 x = \dfrac{\text{adjacent}^2}{\text{hypotenuse}^2}$.
- Step 5 — Add them: $\sin^2 x + \cos^2 x = \dfrac{\text{opposite}^2 + \text{adjacent}^2}{\text{hypotenuse}^2}$.
- Step 6 — By the Pythagorean theorem of right triangles, $\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2$.
- Step 7 — Therefore $\sin^2 x + \cos^2 x = \dfrac{\text{hypotenuse}^2}{\text{hypotenuse}^2} = 1$.
- Step 8 — Both geometric and algebraic reasoning confirm the identity.
- Final Answer: $\boxed{\sin^2 x + \cos^2 x = 1}$ is verified.
FAQs
Radians or degrees?
Symbolic steps are independent; numeric checks use radians by default.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.