problem with proving identities using basic identities

**lindros** · June 22nd 2010, 11:27 PM

i have a problem with this:

prove,
a) $\frac{1-tan^2\theta}{cot^2 \theta-1}=tan^2\theta$

b) $(cos \theta+sin \theta)^2+(cos \theta-sin \theta)^2=2$

any help will appreciate!! thanks.

**Unknown008** · June 22nd 2010, 11:32 PM

a) $\frac{1-tan^2\theta}{cot^2\theta-1} = \frac{1-\frac{sin^2\theta}{cos^2\theta}}{\frac{cos^2\theta }{sin^2\theta} - 1}$

$= \frac{\frac{cos^2\theta - sin^2\theta}{cos^2\theta}}{\frac{cos^2\theta - sin^2\theta}{sin^2\theta}}$

$=\frac{\frac{1}{cos^2\theta}}{\frac{1}{sin^2\theta }}$

$= \frac{sin^2\theta}{cos^2\theta}$

$=tan^2\theta$

**Unknown008** · June 22nd 2010, 11:35 PM

b) $(cos\theta + sin\theta)^2 + (cos\theta - sin\theta)^2 = cos^2 \theta + 2cos\theta sin\theta + sin^2\theta + cos^2\theta -2cos\theta sin\theta + sin^2\theta$

$= cos^2 \theta + \cancel{2cos\theta sin\theta} + sin^2\theta + cos^2\theta \cancel{-2cos\theta sin\theta} + sin^2\theta$

$= 2cos^2\theta + 2 sin^2\theta$

$= 2(cos^2\theta + sin^2\theta)$

$= 2(1)$

$= 2$

**lindros** · June 22nd 2010, 11:50 PM

Originally Posted by Unknown008

a) $\frac{1-tan^2\theta}{cot^2\theta-1} = \frac{1-\frac{sin^2\theta}{cos^2\theta}}{\frac{cos^2\theta }{sin^2\theta} - 1}$

$= \frac{\frac{cos^2\theta - sin^2\theta}{cos^2\theta}}{\frac{cos^2\theta - sin^2\theta}{sin^2\theta}}$

$=\frac{\frac{1}{cos^2\theta}}{\frac{1}{sin^2\theta }}$

$= \frac{sin^2\theta}{cos^2\theta}$

$=tan^2\theta$

i want to know,how can $\frac{cos^2\theta-sin^2\theta}{cos^2 \theta}=\frac{1}{cos^2\theta}$ ??

**Prove It** · June 22nd 2010, 11:55 PM

Because $\frac{\frac{\cos^2{\theta} - \sin^2{\theta}}{\cos^2{\theta}}}{\frac{\cos^2{\the ta} - \sin^2{\theta}}{\sin^2{\theta}}} = \frac{\frac{1}{\cos^2{\theta}}}{\frac{1}{\sin^2{\t heta}}}\left(\frac{\cos^2{\theta} - \sin^2{\theta}}{\cos^2{\theta} - \sin^2{\theta}}\right)$

$= \frac{\frac{1}{\cos^2{\theta}}}{\frac{1}{\sin^2{\t heta}}}$

**Unknown008** · June 22nd 2010, 11:56 PM

Ok, I'll break up the steps, but this is not what I meant.

$= \frac{\frac{cos^2\theta - sin^2\theta}{cos^2\theta}}{\frac{cos^2\theta - sin^2\theta}{sin^2\theta}}$

$= \frac{cos^2\theta - sin^2\theta}{cos^2\theta} \times \frac{sin^2\theta}{cos^2\theta - sin^2\theta}$

Then:

$= \frac{1}{cos^2\theta} \times \frac{sin^2\theta}{1}$

**lindros** · June 22nd 2010, 11:59 PM

thanks very much both of you sir!!

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