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Math  /  Trigonometry

Question1) cos⁡θ×tan⁡θ=sin⁡θ\cos \theta \times \tan \theta=\sin \theta

Studdy Solution

STEP 1

1. We are given the trigonometric equation cos⁡θ×tan⁡θ=sin⁡θ \cos \theta \times \tan \theta = \sin \theta .
2. We need to solve for θ \theta in terms of known trigonometric identities.
3. We assume that θ \theta is in a domain where all trigonometric functions are defined.

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 equation.
3. Simplify the equation.
4. Solve for θ\theta.

STEP 3

Express 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 equation:
cos⁡θ×sin⁡θcos⁡θ=sin⁡θ\cos \theta \times \frac{\sin \theta}{\cos \theta} = \sin \theta

STEP 5

Simplify the equation. Notice that cos⁡θ\cos \theta in the numerator and denominator cancels out:
sin⁡θ=sin⁡θ\sin \theta = \sin \theta
This equation is an identity, meaning it is true for all values of θ\theta where the original functions are defined.

STEP 6

Since the equation simplifies to an identity, θ\theta can be any angle where tan⁡θ\tan \theta and cos⁡θ\cos \theta are defined. Specifically, θ\theta cannot be values where cos⁡θ=0\cos \theta = 0 because tan⁡θ\tan \theta would be undefined. These are values like θ=π2+nπ\theta = \frac{\pi}{2} + n\pi, where nn is an integer.
The solution is that θ\theta can be any angle except where cos⁡θ=0\cos \theta = 0.

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