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QuestionProve the identity sin⁡θsec⁡θcot⁡θ=1\sin \theta \sec \theta \cot \theta = 1 by simplifying the left side using sin⁡θ\sin \theta and cos⁡θ\cos \theta.

Studdy Solution

STEP 1

Assumptions1. sin⁡θ\sin \theta, sec⁡θ\sec \theta, and cot⁡θ\cot \theta are trigonometric functions. . sec⁡θ\sec \theta is the reciprocal of cos⁡θ\cos \theta.
3. cot⁡θ\cot \theta is the reciprocal of tan⁡θ\tan \theta which is sin⁡θcos⁡θ\frac{\sin \theta}{\cos \theta}.
4. The goal is to transform the left side of the equation into the right side.

STEP 2

First, we need to rewrite the trigonometric functions in terms of sin⁡θ\sin \theta and cos⁡θ\cos \theta.
sin⁡θsec⁡θcot⁡θ=sin⁡θ⋅1cos⁡θ⋅cos⁡θsin⁡θ\sin \theta \sec \theta \cot \theta = \sin \theta \cdot \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta}

STEP 3

Now, we can simplify the equation by cancelling out the common terms.
sin⁡θ⋅1cos⁡θ⋅cos⁡θsin⁡θ=sin⁡θ⋅cos⁡θcos⁡θ⋅sin⁡θ\sin \theta \cdot \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} = \frac{\sin \theta \cdot \cos \theta}{\cos \theta \cdot \sin \theta}

STEP 4

Further simplification leads to the right side of the equation.
sin⁡θ⋅cos⁡θcos⁡θ⋅sin⁡θ=1\frac{\sin \theta \cdot \cos \theta}{\cos \theta \cdot \sin \theta} =1Because we have succeeded in transforming the left side into the right side, we have shown that the statement sin⁡θsec⁡θcot⁡θ=1\sin \theta \sec \theta \cot \theta=1 is an identity.

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