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Question: If \(\cos\theta = \frac{3}{5}\)and \(\cos\varphi = \frac{4}{5},\) where \(\theta\)and \(\varphi\)are...

If cos⁡θ=35\cos\theta = \frac{3}{5}and cos⁡φ=45,\cos\varphi = \frac{4}{5}, where θ\thetaand φ\varphiare positive acute angles, then cos⁡θ−φ2=\cos\frac{\theta - \varphi}{2} =

A

72\frac{7}{\sqrt{2}}

B

752\frac{7}{5\sqrt{2}}

C

75\frac{7}{\sqrt{5}}

D

725\frac{7}{2\sqrt{5}}

Answer

752\frac{7}{5\sqrt{2}}

Explanation

Solution

We have cos⁡θ=35\cos\theta = \frac{3}{5}andcos⁡φ=45\cos\varphi = \frac{4}{5}.

Therefore cos⁡(θ−φ)=cos⁡θcos⁡φ+sin⁡θsin⁡φ\cos(\theta - \varphi) = \cos\theta\cos\varphi + \sin\theta\sin\varphi

=35.45+45.35=2425= \frac{3}{5}.\frac{4}{5} + \frac{4}{5}.\frac{3}{5} = \frac{24}{25}

But 2cos⁡2(θ−φ2)=1+cos⁡(θ−φ)=1+2425=49502\cos^{2}\left( \frac{\theta - \varphi}{2} \right) = 1 + \cos(\theta - \varphi) = 1 + \frac{24}{25} = \frac{49}{50}

∴cos⁡2(θ−φ2)=4950\cos^{2}\left( \frac{\theta - \varphi}{2} \right) = \frac{49}{50}. Hence, cos⁡(θ−φ2)=752\cos\left( \frac{\theta - \varphi}{2} \right) = \frac{7}{5\sqrt{2}}.