Prove the identity: \frac{\sin \theta \csc \theta}{\cot \theta} = \tan \theta

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Understand the Problem

The question is asking to prove the given trigonometric identity involving sine, cosecant, and cotangent functions, showing that the left-hand side simplifies to the right-hand side.

Answer

$$ \frac{\sin \theta \csc \theta}{\cot \theta} = \tan \theta $$
Answer for screen readers

The identity is proven as: $$ \frac{\sin \theta \csc \theta}{\cot \theta} = \tan \theta $$

Steps to Solve

  1. Rewrite the functions in basic terms

Start with the left-hand side of the identity: $$ \frac{\sin \theta \csc \theta}{\cot \theta} $$

Recall that:

  • The cosecant function is defined as $ \csc \theta = \frac{1}{\sin \theta} $.
  • The cotangent function is defined as $ \cot \theta = \frac{\cos \theta}{\sin \theta} $.
  1. Substitute the definitions

Substituting the definitions into the expression: $$ \frac{\sin \theta \cdot \frac{1}{\sin \theta}}{\frac{\cos \theta}{\sin \theta}} $$

  1. Simplify the expression

This simplifies to: $$ \frac{1}{\frac{\cos \theta}{\sin \theta}} = \frac{1 \cdot \sin \theta}{\cos \theta} $$

  1. Use basic trigonometric identities

This expression can be rewritten as: $$ \frac{\sin \theta}{\cos \theta} = \tan \theta $$

  1. Conclude the proof

Thus, we have shown that: $$ \frac{\sin \theta \csc \theta}{\cot \theta} = \tan \theta $$

The identity is proven as: $$ \frac{\sin \theta \csc \theta}{\cot \theta} = \tan \theta $$

More Information

This identity is useful in various trigonometry problems and establishing relationships between different trigonometric functions. Understanding these identities helps simplify complex trigonometric expressions in mathematics.

Tips

  • Forgetting to use the definitions of trigonometric functions correctly.
  • Failing to simplify the expression fully, leaving it in a fraction form without recognizing it can reduce further.

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