Starting from $\sin heta = y$, $\cos heta = x$ for the point $(x,y)$ on the unit circle, prove that $1 + an^2 heta = \sec^2 heta$ for every $ heta$ at which $ an heta$ is defined.

Question

Starting from $\sin	heta = y$, $\cos	heta = x$ for the point $(x,y)$ on the unit circle, prove that $1 + 	an^2	heta = \sec^2	heta$ for every $	heta$ at which $	an	heta$ is defined.

Accepted Answer

For the point $(x,y)=(\cos	heta,\sin	heta)$ on the unit circle, $x^2+y^2=1$
(the circle's equation), i.e. $\cos^2	heta+\sin^2	heta=1$ (Section 4). Dividing every term by
$\cos^2	heta$ (valid where $\cos	heta
eq0$, i.e. wherever $	an	heta$ is defined): …

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.Starting from $\sin\theta = y$ , $\cos\theta = x$ for the point $(x,y)$ on the unit circle, prove that $1 + \tan^2\theta = \sec^2\theta$ for every $\theta$ at which $\tan\theta$ is defined.

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.Starting from sin⁡θ=y\sin\theta = ysinθ=y, cos⁡θ=x\cos\theta = xcosθ=x for the point (x,y)(x,y)(x,y) on the unit circle, prove that 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta1+tan2θ=sec2θ for every θ\thetaθ at which tan⁡θ\tan\thetatanθ is defined.

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Q.Starting from $\sin\theta = y$ , $\cos\theta = x$ for the point $(x,y)$ on the unit circle, prove that $1 + \tan^2\theta = \sec^2\theta$ for every $\theta$ at which $\tan\theta$ is defined.

Unlock everything free for 14 days