Question

If$\cos 20°-\sin 20°=p$then $\cos 40°$is equal to

• A. $p^2\sqrt{(2–p^2)}$
• B. $p\sqrt{(2–p^2)}$
• C. $p+\sqrt{(2–p^2)}$
• D. $p–\sqrt{(2–p^2)}$

Answer 1

The correct option is B

$p\sqrt{(2–p^2)}$

The explanation for the correct option :

Find the value of $\cos 40°$:

Given, $\cos 20°–\sin 20°=p$

Squaring on both sides,

$$\begin{gathered} \Rightarrow {\cos }^2(20°)+{\sin }^2(20°)–2\sin 20°\cos 20°=p^2 \\ \Rightarrow 1–\sin 2\times (20°)=p^2 \\ \Rightarrow \sin 40°=1–p^2 \end{gathered}$$

Now,

$\begin{array}{rcl}\cos 40°& =& \sqrt{[1–{\sin }^2(40°\left)\right]}\\ & =& \sqrt{1–{(1–p^2)}^2}\\ & =& \sqrt{[1–(1+p^4–2p^2\left)\right]}\\ & =& \sqrt{(2p^2–p^4)}\\ & =& \sqrt{\left[p^2\right(2–p^2\left)\right]}\\ & =& p\sqrt{(2–p^2)}\end{array}$

Hence, option (B) is the correct answer.