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Apply reciprocal and quotient identities

Reciprocal identities express cosecant, secant, and cotangent as the reciprocals of sine, cosine, and tangent, while quotient identities express tangent as sin⁡θ/cos⁡θ\sin\theta/\cos\theta and cotangent as cos⁡θ/sin⁡θ\cos\theta/\sin\theta, with each equation restricted to angles for which its denominator is nonzero. This understanding supports rewriting, simplifying, and verifying elementary trigonometric expressions and prevents confusing reciprocal functions with inverse trigonometric functions; more advanced identity systems and generalized treatments are not included.

Detailed Explanation: Apply reciprocal and quotient identities

Reciprocal identities rewrite a trigonometric function as 11 divided by another function:

csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ.\csc\theta=\frac{1}{\sin\theta},\qquad \sec\theta=\frac{1}{\cos\theta},\qquad \cot\theta=\frac{1}{\tan\theta}.

Quotient identities rewrite tangent and cotangent using sine and cosine:

tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ.\tan\theta=\frac{\sin\theta}{\cos\theta},\qquad \cot\theta=\frac{\cos\theta}{\sin\theta}.

These equations are used only when their denominators are nonzero. For example, tan⁡θ=sin⁡θcos⁡θ\tan\theta=\frac{\sin\theta}{\cos\theta} requires cos⁡θ≠0\cos\theta\ne0. Reciprocal functions such as sec⁡θ\sec\theta are not the same as inverse trigonometric functions such as arccos⁡θ\arccos\theta.

Worked example

Simplify

tan⁡θsec⁡θ,\frac{\tan\theta}{\sec\theta},

assuming cos⁡θ≠0\cos\theta\ne0.

Step 1: Apply the quotient identity to tangent.

tan⁡θ=sin⁡θcos⁡θ\tan\theta=\frac{\sin\theta}{\cos\theta}

Step 2: Apply the reciprocal identity to secant.

sec⁡θ=1cos⁡θ\sec\theta=\frac{1}{\cos\theta}

Substitute both identities:

tan⁡θsec⁡θ=sin⁡θcos⁡θ1cos⁡θ.\frac{\tan\theta}{\sec\theta} = \frac{\frac{\sin\theta}{\cos\theta}}{\frac{1}{\cos\theta}}.

Step 3: Divide by a fraction by multiplying by its reciprocal.

sin⁡θcos⁡θ1cos⁡θ=sin⁡θcos⁡θ⋅cos⁡θ1.\frac{\frac{\sin\theta}{\cos\theta}}{\frac{1}{\cos\theta}} = \frac{\sin\theta}{\cos\theta}\cdot\frac{\cos\theta}{1}.

Step 4: Cancel cos⁡θ\cos\theta. This is valid because cos⁡θ≠0\cos\theta\ne0.

sin⁡θcos⁡θ⋅cos⁡θ=sin⁡θ.\frac{\sin\theta}{\cos\theta}\cdot\cos\theta = \sin\theta.

Therefore,

tan⁡θsec⁡θ=sin⁡θ\boxed{\frac{\tan\theta}{\sec\theta}=\sin\theta}

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

Learn by doing: Apply reciprocal and quotient identities

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