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7.2 Trigonometric Integrals Trig Identities Basic identities sin 2 x cos2 x 1 tan 2 x 1 sec2 x cot 2 x 1 csc2 x Half-angle identities 1 1 2 2 sin x (1 cos 2 x) cos x (1 cos 2 x) 2 2 Product-to-Sum identities sin A cos B 12 [sin( A B) sin( A B)] cos A cos B 12 [cos( A B) cos( A B)] sin A sin B 12 [cos( A B) cos( A B)] Examples cos 3 4 x sin x dx 6 4 tan x sec x dx 7 4 sin x cos x dx 3 5 tan x sec x dx 4 sin x dx sin( 5 x) cos(3x) dx 6 sin x cos x dx 3 sin x 1 cos x 2 dx Strategy 1) m n sin x cos x dx If one of the power is odd, save one sine (or cosine) factor, and express the remaining expression in terms of cosine (or sine). Then use substitution. If both powers are even, use half-angle identities to reduce the powers 2) m n tan x sec x dx If power of secant is even, save a factor of sec2x and express the remaining expression in terms of tan x. Then use substitution. If power of tangent is odd, save a factor of (sec x tan x) and express the remaining expression in terms of sec x. Then use substitution. 3) sin mx cos nx or sin mx sin nx or Use product-to-sum identities. cos mx cos nx