Trigonometric Identities

**magentarita** · August 23rd 2008, 01:47 PM

Establish each identity below.

(1) 9 sec^2 (x) - 5 tan^2 (x) = 5 + 4 sec^2 (x)

(2) cos(x)/[1 + sin(x)] + [1 + sin(x)]/cos(x) = 2 sec (x)

**o_O** · August 23rd 2008, 01:53 PM

#1:
Note that $\tan^{2} x = \sec^{2} x - 1$ . So substitute it in and simplify.

#2:
Combine into a single fraction:
$\frac{\cos x}{1+\sin x} {\color{blue}\cdot \frac{\cos x}{\cos x} } \: + \: \frac{1+ \sin x}{\cos x} {\color{blue} \cdot \frac{1+ \sin x}{1 + \sin x} }$

$= \frac{\cos^2 x + (1+ \sin x)^2}{\cos x(1+ \sin x)}$

Expand the numerator and simplify.

Post back if you have any troubles!

**magentarita** · August 23rd 2008, 03:26 PM

Originally Posted by o_O

#1:
Note that $\tan^{2} x = \sec^{2} x - 1$ . So substitute it in and simplify.

#2:
Combine into a single fraction:
$\frac{\cos x}{1+\sin x} {\color{blue}\cdot \frac{\cos x}{\cos x} } \: + \: \frac{1+ \sin x}{\cos x} {\color{blue} \cdot \frac{1+ \sin x}{1 + \sin x} }$

$= \frac{\cos^2 x + (1+ \sin x)^2}{\cos x(1+ \sin x)}$

Expand the numerator and simplify.

Post back if you have any troubles!

What in the question indicated that you needed to make cosx/cosx
on the left side and 1 + sinx/1 + sinx on the right side?

**skeeter** · August 23rd 2008, 04:08 PM

What in the question indicated that you needed to make cosx/cosx
on the left side and 1 + sinx/1 + sinx on the right side?

basic addition of fractions ... you need the common denominator, $\cos{x}(1 + \sin{x})$ , to add the two fractions.

**magentarita** · August 23rd 2008, 05:51 PM

Originally Posted by skeeter

basic addition of fractions ... you need the common denominator, $\cos{x}(1 + \sin{x})$ , to add the two fractions.

It's searching for the LCD, right?

**o_O** · August 23rd 2008, 06:52 PM

An analogy:

$\frac{1}{2} + \frac{2}{3}$

$= \frac{1}{2} {\color{blue}\cdot \frac{3}{3} } + \frac{2}{3} {\color{blue}\cdot \frac{2}{2}}$

$= \frac{3}{6} + \frac{4}{6}$

$=\frac{7}{6}$

**magentarita** · August 24th 2008, 02:48 PM

Great analogy!

Thanks

Thread: Trigonometric Identities

LinkBack

Thread Tools

Display