Search any question & find its solution
Question:
Answered & Verified by Expert
If $\mathrm{x}=\sin 70^{\circ} \sin 50^{\circ}$ and $\mathrm{y}=\cos 60^{\circ} \cos 80^{\circ}$, then what is $\mathrm{xy}$
equal to?
Options:
equal to?
Solution:
2243 Upvotes
Verified Answer
The correct answer is:
$1 / 16$
$x=\sin 70^{\circ} \cdot \sin 50^{\circ}$ and $y=\cos 60^{\circ} \cdot \cos 80^{\circ}$
$\begin{aligned} \Rightarrow x y=& \cos 60^{\circ} \cdot \sin 70^{\circ} \cdot \sin 50^{\circ} \cdot \cos 80^{\circ} \\ x y=\frac{1}{2} \cdot & \sin (90-20) \cdot \sin (90-40) \cdot \cos 80 \\ \Rightarrow x y=& \frac{1}{2} \cdot \cos 20 \cdot \cos 40 \cdot \cos 80 \\(\because \sin (90-x)=\cos x) \\ \Rightarrow x y=& \frac{1}{2} \cdot \cos 20^{\circ} \cdot \cos (60-20)^{\circ} \cdot \cos (60+20)^{\circ} \\ \Rightarrow x y=& \frac{1}{2}\left[\frac{1}{4} \cos 3\left(20^{\circ}\right)\right]=\frac{1}{2} \times \frac{1}{4} \times \cos 60^{\circ}=\frac{1}{16} . \\ &\left[\because \cos \theta \cdot \cos (60-\theta) \cdot \cos (60+\theta)=\frac{1}{4} \cos 3 \theta\right] \\
$\begin{aligned} \Rightarrow x y=& \cos 60^{\circ} \cdot \sin 70^{\circ} \cdot \sin 50^{\circ} \cdot \cos 80^{\circ} \\ x y=\frac{1}{2} \cdot & \sin (90-20) \cdot \sin (90-40) \cdot \cos 80 \\ \Rightarrow x y=& \frac{1}{2} \cdot \cos 20 \cdot \cos 40 \cdot \cos 80 \\(\because \sin (90-x)=\cos x) \\ \Rightarrow x y=& \frac{1}{2} \cdot \cos 20^{\circ} \cdot \cos (60-20)^{\circ} \cdot \cos (60+20)^{\circ} \\ \Rightarrow x y=& \frac{1}{2}\left[\frac{1}{4} \cos 3\left(20^{\circ}\right)\right]=\frac{1}{2} \times \frac{1}{4} \times \cos 60^{\circ}=\frac{1}{16} . \\ &\left[\because \cos \theta \cdot \cos (60-\theta) \cdot \cos (60+\theta)=\frac{1}{4} \cos 3 \theta\right] \\
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.