How to verify the trigonometric identity tanθ cosθ = sinθ

Key concept: When you see a trig identity like this, the fastest path is usually to rewrite everything in sine and cosine. That keeps the algebra honest and makes cancellations visible.

Step by step

Step 1: Rewrite $\tan\theta$ in terms of sine and cosine

Use the identity $\tan\theta=\frac{\sin\theta}{\cos\theta}.$

Substitute it into the left-hand side: $\tan\theta\cos\theta=\left(\frac{\sin\theta}{\cos\theta}\right)\cos\theta.$

Step 2: Cancel the common factor

The $\cos\theta$ in the numerator and denominator cancels: $\left(\frac{\sin\theta}{\cos\theta}\right)\cos\theta=\sin\theta.$

So the left-hand side simplifies to the right-hand side.

Step 3: State the conclusion

Therefore, $\tan\theta\cos\theta=\sin\theta,$ so the identity is verified.

Pitfall alert

A common mistake is trying to prove identities by plugging in random angle values. That can check a case, but it does not verify the identity. Also remember that $\tan\theta$ is undefined when $\cos\theta=0$, so the identity is understood where both sides are defined.

Try different conditions

If the problem were written as $\sin\theta\cdot\sec\theta=\tan\theta$, you would use $\sec\theta=\frac{1}{\cos\theta}$ instead. That gives $\sin\theta\sec\theta=\sin\theta\cdot\frac{1}{\cos\theta}=\frac{\sin\theta}{\cos\theta}=\tan\theta.$

Further reading

trigonometric identity, sine and cosine, quotient identity