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user100 [1]
3 years ago
9

Sin(60-theta)sin(60+theta)​

Mathematics
2 answers:
nexus9112 [7]3 years ago
5 0

Answer:

It is a double angle of cosine

ehidna [41]3 years ago
3 0

Step-by-step explanation:

\sin(60 -  \theta)  \sin(60 +  \theta)  \\  =  \{ \sin(60)  \cos( \theta)  -  \sin( \theta)   \cos(60)  \} \{ \sin(60)  \cos( \theta)  +  \cos(60)  \sin( \theta)  \} \\  =  \{ \frac{ \sqrt{3} }{2}  \cos( \theta)  -  \frac{1}{2}  \sin( \theta)  \} \{ \frac{ \sqrt{3} }{2}  \cos( \theta)  +  \frac{1}{2}  \sin( \theta)  \} \\  \\

from difference of two squares:

{ \boxed{(a - b)(a + b) = ( {a}^{2} -  {b}^{2} ) }}

therefore:

=  \{ {( \frac{ \sqrt{3} }{2}) }^{2}  { \cos }^{2}  \theta \} -  \{ {( \frac{ \sqrt{3} }{2} )}^{2}  { \sin }^{2}  \theta \} \\  \\  =  \frac{3}{4}  { \cos }^{2}  \theta -  \frac{3}{4}  { \sin}^{2}  \theta

factorise out ¾ :

=  \frac{3}{4} ( { \cos }^{2}  \theta  -  { \sin}^{2}   \theta) \\  \\  = { \boxed{ \frac{3}{4}  \cos(2 \theta) }}

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