Question

Verify the identity. \(\frac{\cos \theta \tan \theta+\sin \theta}{\tan \theta}=2 \cos \theta\)

Short answer

To verify the given trigonometric identity \(\frac{\cos \theta \tan \theta+\sin \theta}{\tan \theta}=2 \cos \theta\), first rewrite all trigonometric functions in terms of sine and cosine, then simplify and combine terms to obtain the simplified expression \(2 \cos \theta\). Since the simplified expression is equal to the right-hand side of the equation, we have verified the given identity.

Step-by-step solution

Rewrite the Trigonometric Functions

We will first express all trigonometric functions in terms of sine and cosine functions: $$\frac{\cos \theta \tan \theta+\sin \theta}{\tan \theta}=\frac{\cos \theta \cdot\frac{\sin \theta}{\cos \theta}+\sin \theta}{\frac{\sin \theta}{\cos \theta}}$$

Simplify the Numerator

Now, we will simplify the numerator by cancelling out the common terms (\(\cos\theta\)): $$=\frac{\sin \theta+\sin \theta}{\frac{\sin \theta}{\cos \theta}}$$

Combine the Sine Terms

Next, we combine the sine terms in the numerator: $$=\frac{2\sin \theta}{\frac{\sin \theta}{\cos \theta}}$$

Multiply by the Reciprocal

Now, we will multiply the fraction by the reciprocal of the denominator to eliminate the fraction in the denominator: $$=2\sin \theta\cdot\frac{\cos \theta}{\sin \theta}$$

Cancel the Common Terms and Simplify

Cancel the common terms(\(\sin\theta\)) and simplify the expression: $$=2 \cos \theta$$ Since we arrived at the RHS, we have verified the given identity: $$\frac{\cos \theta \tan \theta+\sin \theta}{\tan \theta}=2 \cos \theta$$