Question

Find the value of $\cos \theta -\cos 3\theta +\cos 4\theta$, when $\theta ={45}^{\circ }$.

• A. $\frac{\sqrt{3}}{2}$
• B. $\sqrt{3}-1$
• C. $\frac{1}{2}$
• D. $\sqrt{2}-1$

Answer 1

The correct option is D $\sqrt{2}-1$

Given exp. = cosθ−cos3θ+cos4θcos⁡θ−cos⁡3θ+cos⁡4θ
=2sin2θsinθ+cos4θ$=2sin⁡2θsin⁡θ+cos⁡4θ
[∴θ=45∘][∴θ=45∘]
=2sin90∘sin45∘+cos180∘=2sin⁡90∘sin⁡45∘+cos⁡180∘
=2×1×12√−1=2√−1=2×1×12−1=2−1