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Prove the trigonometric identity: 2 theta - 2 theta = 2theta

Mathematics

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Step-by-step solution

3 steps

Here's the solution for question 5(a).

  1. Start with the left-hand side (LHS) of the identity.

    LHS=cos⁡2θ−sin⁡2θ\text{LHS} = \cos^2 \theta - \sin^2 \theta

  2. Apply the double angle identity for cosine.

    The fundamental trigonometric identity for cos⁡2θ\cos 2\theta is:

    cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta = \cos^2 \theta - \sin^2 \theta

  3. Conclude that the LHS is equal to the right-hand side (RHS).

    From Step 2, we can directly substitute cos⁡2θ−sin⁡2θ\cos^2 \theta - \sin^2 \theta with cos⁡2θ\cos 2\theta.

    LHS=cos⁡2θ\text{LHS} = \cos 2\theta LHS=RHS\text{LHS} = \text{RHS}

    The identity is proven.

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