How to simplify this Cosecant identities?

How to find the value of this equation? I've tried about the multiplication and addition of sine and cosine, but it still useless, and I always ended up with the sin 10 at the end.

#Geometry #HelpMe! #MathProblem

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Recall the identities

$\sin (A) + \sin (B) = 2 \sin ( \frac {A+B}{2} ) \space \cos ( \frac {A-B}{2} )$

$- \cos (A) + \cos (B) = 2 \sin ( \frac {A+B}{2} ) \space \sin ( \frac {A-B}{2} )$

$\cos (A) + \cos (B) = 2 \cos ( \frac {A+B}{2} ) \space \cos ( \frac {A-B}{2} )$

We have,

$\large \frac {1}{\sin (10^\circ )} + \frac {1}{\sin (50^\circ )} - \frac {1}{\sin (70^\circ )}$

$\large = \frac {1}{\sin (10^\circ )} + \frac {1}{\sin (50^\circ )} - \frac {1}{\cos (20^\circ )}$ , because $\sin (A) = \cos (90^\circ - A)$

$\large = \frac {2 ( \space \sin (50^\circ ) + \sin (10^\circ ) \space ) } { 2 \sin (50^\circ ) \space \sin (10^\circ ) } - \frac {1}{\cos (20^\circ )}$

$\large = \frac {2 (2 \sin (30^\circ) \cos (20^\circ) ) } { - \cos (60^\circ) + \cos (40^\circ) } - \frac {1}{\cos (20^\circ )}$

$\large = \frac {4 \cos (20^\circ) } { - 1 + 2\cos (40^\circ) } - \frac {1}{\cos (20^\circ )}$

$\large = \frac {4 \cos^2 (20^\circ) -2 \cos (40^\circ) + 1} { 2 \cos (40^\circ) \space \cos (20^\circ) - \cos (20^\circ) }$

$\large = \frac {4 \cos^2 (20^\circ) - 2 -2 \cos (40^\circ) + 3} { ( \space \cos (60^\circ) + \cos (20^\circ) ) - \cos (20^\circ) }$

$\large = \frac { 2( 2 \cos^2 (20^\circ) - 1 ) -2 \cos (40^\circ) + 3} { \cos (60^\circ) }$ , because we want to apply $\cos (2A) = 2 \cos^2 (A) - 1$

$\large = \frac { 2 \cos (40^\circ) -2 \cos (40^\circ) + 3} { \frac {1}{2} }$

$\large = 3 \times 2 = 6$

Pi Han Goh - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$