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Verify fundamental trigonometric identities

Verification of fundamental trigonometric identities involves recognizing that relationships such as sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1, reciprocal identities, and quotient identities hold for every real angle for which both sides are defined. Algebraic reasoning—factoring, finding common denominators, and substituting known identities—establishes equivalence by transforming one side into the other, while distinguishing identities from equations true only for particular angles and tracking denominator restrictions; more advanced identities involving complex numbers or generalized proof frameworks are outside this scope.

Detailed Explanation: Verify fundamental trigonometric identities

To verify a trigonometric identity, transform one side only until it becomes the other side. Use known identities, algebraic operations, and common denominators. Also check where the expressions are defined.

Useful fundamental identities include

sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1 sec⁡θ=1cos⁡θandtan⁡θ=sin⁡θcos⁡θ.\sec\theta=\frac{1}{\cos\theta} \qquad\text{and}\qquad \tan\theta=\frac{\sin\theta}{\cos\theta}.

Example

Verify that

sec⁡θ−cos⁡θ=sin⁡2θcos⁡θ.\sec\theta-\cos\theta=\frac{\sin^2\theta}{\cos\theta}.

First, note the denominator restriction:

cos⁡θ≠0.\cos\theta\ne 0.

This is required because both sec⁡θ\sec\theta and sin⁡2θcos⁡θ\dfrac{\sin^2\theta}{\cos\theta} are undefined when cos⁡θ=0\cos\theta=0.

Start with the left-hand side:

sec⁡θ−cos⁡θ\sec\theta-\cos\theta

Use the reciprocal identity sec⁡θ=1cos⁡θ\sec\theta=\dfrac{1}{\cos\theta}:

1cos⁡θ−cos⁡θ\frac{1}{\cos\theta}-\cos\theta

Rewrite cos⁡θ\cos\theta with denominator cos⁡θ\cos\theta:

1cos⁡θ−cos⁡2θcos⁡θ\frac{1}{\cos\theta}-\frac{\cos^2\theta}{\cos\theta}

Combine the fractions:

1−cos⁡2θcos⁡θ\frac{1-\cos^2\theta}{\cos\theta}

Using sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1, rewrite 1−cos⁡2θ1-\cos^2\theta as sin⁡2θ\sin^2\theta:

sin⁡2θcos⁡θ\frac{\sin^2\theta}{\cos\theta}

This matches the right-hand side, so

sec⁡θ−cos⁡θ=sin⁡2θcos⁡θ\boxed{\sec\theta-\cos\theta=\frac{\sin^2\theta}{\cos\theta}}

for all angles where cos⁡θ≠0\cos\theta\ne0.

An identity is true for every angle in its domain; it is not something that needs to be solved for particular angle values.

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    Trigonometry Identities - Pythagorean (Tan^2 and Sin^2/Cos^2) Identity True/False (Greek Letter)

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