2.3 Trigonometry

cosec x=1/sin x, sec x=1/cos x and cot x=1/tan x=cos x/sin x wherever the denominator is non-zero. Their graphs inherit zeros and asymptotes from the original functions.

Use reciprocal values rather than inventing new triangle rules. Mark undefined angles and use the original sine or cosine sign to determine the reciprocal sign.

sec(π/3)=2 because cos(π/3)=1/2; sec x is undefined where cos x=0.

cosec x is not sin⁻¹x, and reciprocal graphs do not cross the x-axis because a reciprocal cannot equal zero.

Use compound-angle, double-angle and half-angle identities to rewrite expressions into a form suited to the question. For example, sin2x=2sinx cosx and cos2x=1−2sin²x.

Choose one direction of simplification, keep domain restrictions, and avoid replacing an identity with a numerical check at a few angles.

1−cos2x=2sin²x converts a cosine expression into a square that is easier to integrate or solve.

An identity is true for every allowed x; a relation that works only at selected angles is not an identity.