Math  /  Trigonometry

QuestionExpress each of the following as a single trigonometric ratio. a) 2sin5xcos5x2 \sin 5 x \cos 5 x d) 2tan4x1tan24x\frac{2 \tan 4 x}{1-\tan ^{2} 4 x} b) cos2θsin2θ\cos ^{2} \theta-\sin ^{2} \theta e) 4sinθcosθ4 \sin ^{\prime} \theta \cos \theta c) 12sin23x1-2 \sin ^{2} 3 x f) 2cos2θ212 \cos ^{2} \frac{\theta}{2}-1

Studdy Solution

STEP 1

1. We will use trigonometric identities to simplify each expression.
2. Each expression can be expressed as a single trigonometric ratio.

STEP 2

1. Simplify expression (a) using a product-to-sum identity.
2. Simplify expression (b) using a difference of squares identity.
3. Simplify expression (c) using a double angle identity.
4. Simplify expression (d) using a tangent double angle identity.
5. Simplify expression (e) using a double angle identity.
6. Simplify expression (f) using a half-angle identity.

STEP 3

For expression (a) 2sin5xcos5x 2 \sin 5x \cos 5x , use the double angle identity for sine:
2sinAcosA=sin2A 2 \sin A \cos A = \sin 2A
Thus,
2sin5xcos5x=sin10x 2 \sin 5x \cos 5x = \sin 10x

STEP 4

For expression (b) cos2θsin2θ \cos^2 \theta - \sin^2 \theta , use the identity:
cos2Asin2A=cos2A \cos^2 A - \sin^2 A = \cos 2A
Thus,
cos2θsin2θ=cos2θ \cos^2 \theta - \sin^2 \theta = \cos 2\theta

STEP 5

For expression (c) 12sin23x 1 - 2 \sin^2 3x , use the identity:
12sin2A=cos2A 1 - 2 \sin^2 A = \cos 2A
Thus,
12sin23x=cos6x 1 - 2 \sin^2 3x = \cos 6x

STEP 6

For expression (d) 2tan4x1tan24x \frac{2 \tan 4x}{1 - \tan^2 4x} , use the tangent double angle identity:
2tanA1tan2A=tan2A \frac{2 \tan A}{1 - \tan^2 A} = \tan 2A
Thus,
2tan4x1tan24x=tan8x \frac{2 \tan 4x}{1 - \tan^2 4x} = \tan 8x

STEP 7

For expression (e) 4sinθcosθ 4 \sin \theta \cos \theta , use the double angle identity for sine:
2sinAcosA=sin2A 2 \sin A \cos A = \sin 2A
Thus,
4sinθcosθ=2sin2θ 4 \sin \theta \cos \theta = 2 \sin 2\theta

STEP 8

For expression (f) 2cos2θ21 2 \cos^2 \frac{\theta}{2} - 1 , use the identity:
2cos2A1=cos2A 2 \cos^2 A - 1 = \cos 2A
Thus,
2cos2θ21=cosθ 2 \cos^2 \frac{\theta}{2} - 1 = \cos \theta

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