Studdy AI Logo
Math  /  Trigonometry

Question6. Prove the Trig Identities. a. cos⁡θ×tan⁡θ=sin⁡θ\cos \theta \times \tan \theta=\sin \theta

Studdy Solution

STEP 1

1. We assume the basic trigonometric identities are known: tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.
2. We assume the domain of θ\theta is such that cos⁡θ≠0\cos \theta \neq 0.

STEP 2

1. Express tan⁡θ\tan \theta in terms of sin⁡θ\sin \theta and cos⁡θ\cos \theta.
2. Substitute the expression for tan⁡θ\tan \theta into the left-hand side of the identity.
3. Simplify the expression to show it equals the right-hand side.

STEP 3

Start by expressing tan⁡θ\tan \theta in terms of sin⁡θ\sin \theta and cos⁡θ\cos \theta:
tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

STEP 4

Substitute the expression for tan⁡θ\tan \theta into the left-hand side of the identity cos⁡θ×tan⁡θ\cos \theta \times \tan \theta:
cos⁡θ×tan⁡θ=cos⁡θ×sin⁡θcos⁡θ\cos \theta \times \tan \theta = \cos \theta \times \frac{\sin \theta}{\cos \theta}

STEP 5

Simplify the expression by canceling cos⁡θ\cos \theta in the numerator and the denominator:
cos⁡θ×sin⁡θcos⁡θ=sin⁡θ\cos \theta \times \frac{\sin \theta}{\cos \theta} = \sin \theta
The identity is proven:
cos⁡θ×tan⁡θ=sin⁡θ\cos \theta \times \tan \theta = \sin \theta

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord