Trigonometric Proofs

Transactional proofs: Here, trigonometric identities are leveraged as building blocks to create complex statements. Hence, understanding fundamental identities is crucial.
Proof by contradiction: This involves assuming the opposite of what is to be proved, leading to an absurd or impossible result, thus confirming the initial statement is true.
Using Pythagoras’ Theorem to prove trigonometric identities: Various trigonometric proofs are based on the Pythagorean Theorem, mainly when it comes to sine, cosine, and tangent functions.
Composite Function Proofs: Understanding and proving how functions of sine, cosine or tangent combine or relate to each other.

R Addition Formulas

Cosine Addition Formula: Given cos(A+B) and cos(A-B) formulas, you can add or subtract them to find the formulas for cos2A or cos2B.
Sine Addition Formula: Similarly, the sine addition formula can be leveraged to gather more equations for sin2A or sin2B.

Sine Double Angle: The formula sin2A = 2sinAcosA represents the double angle identity of sine.
Cosine Double Angle: There are three variations of cosine double angle identities, which can be derived from the cosine addition formula. They are cos2A = cos^2(A) - sin^2(A), cos2A = 2cos^2(A) - 1, and cos2A = 1 - 2sin^2(A).
Tangent Double Angle: Likewise, the tangent double angle formula, tan2A = 2tanA / (1 - tan^2A), can be found by dividing the sine double angle formula by the cosine double angle formula.

Double Angle: sin2θ, cos2θ and tan2θ should be memorized since they recur frequently.
Co-functions: Knowing the relationship between sine and cosine, tangent and cotangent and so on, and how they change when offsets of ±90° or 180° are involved is beneficial.
Small Angle Approximations: For small angles, sinθ = θ, cosθ = 1, and tanθ = θ are useful approximations. Remember and understand where they come from, i.e., looking at the gradient at θ = 0 on each curve.

Using identities: Knowing identities such as sin2x = 2sin x cosx can make seemingly complex trigonometric equations easily solvable.
General solutions: If you are asked for the general solution, remember to add n360 or n2π to your solution.
Multiple angles: When dealing with equations where trigonometric functions have arguments of multiple angles (like 2x or 3x), remember to solve for the smallest angle first.
Trig equations with ranges: Pay close attention to the range given in the question. Ensure your solutions fall within this range, or adjust appropriately if they do not.