Prove that each equation is an identity. tan((π)/(4)+θ)-tan((π)/(4)-θ)=2 tan2 θHint: Use the add

step 1 In this question we need to evaluate the given expression. 10 pi by 4 plus theta will be 10 pi b

Prove that each equation is an identity....

Prove that each equation is an identity.
$\tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)=2 \tan 2 \theta$
Hint: Use the addition formulas for tangent and the result in Exercise 57

Key Concepts

Trigonometric Identity Verification

Verifying trigonometric identities involves proving that two seemingly different expressions are mathematically equivalent for all values within their domains. This technique requires the application of various trigonometric formulas, algebraic manipulation, and sometimes substitution methods. By breaking down complex expressions into simpler components using known identities, one can systematically demonstrate the equivalence, reinforcing the interconnected nature of trigonometric functions.

Double Angle Formula for Tangent

The double angle formula for tangent relates tan(2?) to tan ?, typically given by tan(2?) = (2 tan ?) / (1 - tan²?). This formula is important in trigonometry because it allows expressions involving angles doubled to be rewritten in terms of single angles, which can simplify proofs and manipulations of identities involving tangent functions.

Tangent Addition Formula

The tangent addition formula expresses tan(A + B) in terms of tan A and tan B. It is given by tan(A + B) = (tan A + tan B) / (1 - tan A tan B). This formula is a fundamental tool in trigonometry that helps in breaking down and simplifying expressions involving the tangent of sum or difference angles, thereby aiding in the derivation and manipulation of various trigonometric identities.

Tangent Subtraction Formula

The tangent subtraction formula is closely related to the addition formula and is used to express tan(A - B) as (tan A - tan B) / (1 + tan A tan B). Like its counterpart, it is vital in simplifying expressions where the difference of angles appears inside a tangent function. Understanding this formula is critical for proving identities that involve both sum and difference forms of trigonometric functions.

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