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Math  /  Trigonometry

Question1. Factor the lollowing a) 4sin⁡2θ−cos⁡2θ4 \sin ^{2} \theta-\cos ^{2} \theta

Studdy Solution

STEP 1

1. We are asked to factor the expression 4sin⁡2θ−cos⁡2θ4 \sin^2 \theta - \cos^2 \theta.
2. The expression involves trigonometric identities that can be used to simplify it.

STEP 2

1. Use trigonometric identities to rewrite the expression.
2. Factor the resulting expression.

STEP 3

Recall the Pythagorean identity:
sin⁡2θ+cos⁡2θ=1 \sin^2 \theta + \cos^2 \theta = 1
This identity can help us express sin⁡2θ\sin^2 \theta in terms of cos⁡2θ\cos^2 \theta or vice versa.

STEP 4

Rewrite 4sin⁡2θ−cos⁡2θ4 \sin^2 \theta - \cos^2 \theta using the identity. We can express sin⁡2θ\sin^2 \theta as:
sin⁡2θ=1−cos⁡2θ \sin^2 \theta = 1 - \cos^2 \theta
Substitute this into the expression:
4(1−cos⁡2θ)−cos⁡2θ 4(1 - \cos^2 \theta) - \cos^2 \theta

STEP 5

Simplify the expression:
4−4cos⁡2θ−cos⁡2θ 4 - 4\cos^2 \theta - \cos^2 \theta
Combine like terms:
4−5cos⁡2θ 4 - 5\cos^2 \theta

STEP 6

Factor the expression 4−5cos⁡2θ4 - 5\cos^2 \theta. Notice that it is already in a simplified form and does not factor further over the real numbers without using complex numbers or additional identities.
Thus, the expression 4sin⁡2θ−cos⁡2θ4 \sin^2 \theta - \cos^2 \theta simplifies to:
4−5cos⁡2θ 4 - 5\cos^2 \theta

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