Proving the Trigonometric Identity ( 1 tan A tan frac{A}{2} sec A )

Trigonometry often involves proving various identities to simplify or solve complex problems. One such identity is ( 1 tan A tan frac{A}{2} sec A ). In this article, we will walk through a step-by-step process of proving this identity using trigonometric identities and relationships. This guide is particularly useful for students and professionals dealing with advanced mathematics and SEO optimization of content.

Step-by-Step Proof

To prove the identity ( 1 tan A tan frac{A}{2} sec A ), we start with the half-angle identity for tangent and proceed through a series of simplifications. Let's break down the proof into detailed steps.

Step 1: Use the Half-Angle Identity for Tangent

The half-angle identity for tangent states:

( tan frac{A}{2} frac{sin A}{1 - cos A} )

Step 2: Substitute ( tan A )

We know that:

( tan A frac{sin A}{cos A} )

Step 3: Substitute and Rewrite the Left-Hand Side

Substituting ( tan A ) and ( tan frac{A}{2} ) into the left-hand side of the equation:

( 1 tan A tan frac{A}{2} 1 left(frac{sin A}{cos A}right) left(frac{sin A}{1 - cos A}right) )

Step 4: Simplify the Expression

To simplify the expression, we combine the fractions:

( 1 frac{sin^2 A}{cos A (1 - cos A)} )

Combining the terms with a common denominator:

( 1 frac{sin^2 A}{cos A (1 - cos A)} frac{cos A (1 - cos A) sin^2 A}{cos A (1 - cos A)} )

Step 5: Use Pythagorean Identity

The Pythagorean identity states:

( sin^2 A cos^2 A 1 )

Substituting ( sin^2 A 1 - cos^2 A ) into the equation:

( cos A (1 - cos A) (1 - cos^2 A) cos A - cos^2 A 1 - cos^2 A cos A 1 - 2 cos^2 A )

Further simplifying:

( cos A 1 - 2 cos^2 A )

This expression can be simplified further by recognizing that the numerator is a simplified form of the identity:

( cos A (1 - cos A) (1 - cos^2 A) cos A (1 - cos A) (1 - cos^2 A) cos A (1 - cos A) sin^2 A )

Step 6: Substitute Back

Now, substitute back into the expression:

( frac{cos A (1 - cos A) sin^2 A}{cos A (1 - cos A)} frac{cos A (1 - cos A) (1 - cos^2 A)}{cos A (1 - cos A)} frac{cos A (1 - cos A) sin^2 A}{cos A (1 - cos A)} )

This simplifies to:

( frac{cos A (1 - cos A) (1 - cos^2 A)}{cos A (1 - cos A)} frac{cos A (1 - cos A) sin^2 A}{cos A (1 - cos A)} frac{1 - cos^2 A}{cos A (1 - cos A)} frac{1 - cos^2 A}{cos A (1 - cos A)} frac{1}{cos A} )

Finally:

( frac{1}{cos A} sec A )

Conclusion

We have shown that:

( 1 tan A tan frac{A}{2} sec A )

This completes the proof, demonstrating the validity of the identity.

Verification

To verify, let's substitute ( A 0^circ ) into the equation:

( 1 tan 0^circ tan frac{0^circ}{2} 1 0 cdot 0 1 )

On the right-hand side:

( sec 0^circ frac{1}{cos 0^circ} frac{1}{1} 1 )

Thus, the left-hand side equals the right-hand side, confirming the identity.