E: Trigonometry: Understand and use sin²θ + cos²θ ≡ 1; sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ.

The idea

This topic covers three fundamental Pythagorean identities in trigonometry. The first, sin²θ + cos²θ ≡ 1, is derived from the unit circle and holds for all θ. Dividing this identity by cos²θ gives sec²θ ≡ 1 + tan²θ, valid where cos θ ≠ 0. Dividing by sin²θ gives cosec²θ ≡ 1 + cot²θ, valid where sin θ ≠ 0. You must understand how these identities are derived and use them to simplify expressions, prove other identities, and solve trigonometric equations. For example, you can replace 1 + tan²θ with sec²θ to simplify an expression, or use sin²θ = 1 - cos²θ to convert between functions. These identities are essential tools for manipulating trigonometric expressions and are frequently examined in proofs and equation solving.

What earns the marks

• States the three Pythagorean identities correctly: sin²θ + cos²θ ≡ 1, sec²θ ≡ 1 + tan²θ, cosec²θ ≡ 1 + cot²θ.
• Derives sec²θ ≡ 1 + tan²θ by dividing sin²θ + cos²θ ≡ 1 by cos²θ, and similarly for cosec²θ and cot²θ.
• Uses the identities to simplify expressions, for example replacing 1 + tan²θ with sec²θ or 1 - sin²θ with cos²θ.
• Solves trigonometric equations by applying the identities to convert between functions, such as solving 2sec²θ = 3 + tan θ.
• Proves other identities by manipulating one side to match the other using these fundamental identities.

Common mistakes and how to put them right

• Writing sin²θ + cos²θ ≡ 1 as sin θ + cos θ ≡ 1, confusing the square of the function with the function itself.

  Put it right: The identity involves the squares of sine and cosine, not the functions themselves.

• Forgetting the restrictions on θ when using sec²θ ≡ 1 + tan²θ or cosec²θ ≡ 1 + cot²θ.

  Put it right: These identities are only valid where the functions involved are defined, i.e., cos θ ≠ 0 for sec and tan, and sin θ ≠ 0 for cosec and cot.

• Misapplying the identities by adding or subtracting terms incorrectly, such as writing sec²θ ≡ 1 - tan²θ.

  Put it right: Sec²θ ≡ 1 + tan²θ, with a plus sign.

Exam tips

• When proving an identity, start with the more complicated side and simplify using the Pythagorean identities until it matches the other side.
• Use the sec²θ and cosec²θ forms directly when they appear; for example, replace 1 + tan²θ with sec²θ rather than converting everything to sin and cos.
• Check for extraneous solutions when solving equations that involve squaring or when the original equation has restrictions on the domain.