Question

simplify: tan^2(x) cos^2(x)+ cot^2(x) sin^2(x)

Answer 1

tan^2x= sin2x/cos2x

cot2x= cos2x/sin2x

sin2x+cos2x=1

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Answer 2

Answer:

$\tan ^2\left(x\right)\cos ^2\left(x\right)+\cot ^2\left(x\right)\sin ^2\left(x\right)=1$

Step-by-step explanation:

$\tan ^2\left(x\right)\cos ^2\left(x\right)+\cot ^2\left(x\right)\sin ^2\left(x\right)$

$\mathrm{Express\:with\:sin,\:cos}$

$\cos ^2\left(x\right)\tan ^2\left(x\right)+\cot ^2\left(x\right)\sin ^2\left(x\right)$

$\mathrm{Using\:the\:Basic\:Trigonometric\:identity}:\quad \tan \left(x\right)=\dfrac{\sin \left(x\right)}{\cos \left(x\right)}$

$\tan \left(x\right)=\dfrac{\sin \left(x\right)}{\cos \left(x\right)}$

$=\cos ^2\left(x\right)\left(\dfrac{\sin \left(x\right)}{\cos \left(x\right)}\right)^2+\cot ^2\left(x\right)\sin ^2\left(x\right)$

$\mathrm{Using\:the\:Basic\:Trigonometric\:identity}:\quad \cot \left(x\right)=\dfrac{\cos \left(x\right)}{\sin \left(x\right)}$

$\cot \left(x\right)=\dfrac{\cos \left(x\right)}{\sin \left(x\right)}$

$=\cos ^2\left(x\right)\left(\dfrac{\sin \left(x\right)}{\cos \left(x\right)}\right)^2+\left(\dfrac{\cos \left(x\right)}{\sin \left(x\right)}\right)^2\sin ^2\left(x\right)$

$=\left(\dfrac{\cos \left(x\right)}{\sin \left(x\right)}\right)^2\sin ^2\left(x\right)+\left(\dfrac{\sin \left(x\right)}{\cos \left(x\right)}\right)^2\cos ^2\left(x\right)$

$\left(\dfrac{\cos \left(x\right)}{\sin \left(x\right)}\right)^2\sin ^2\left(x\right)$

$\left(\dfrac{\cos \left(x\right)}{\sin \left(x\right)}\right)^2$

$\mathrm{Apply\:exponent\:rule}:\quad \left(\dfrac{a}{b}\right)^c=\dfrac{a^c}{b^c}$

$=\dfrac{\cos ^2\left(x\right)}{\sin ^2\left(x\right)}$

$=\dfrac{\cos ^2\left(x\right)}{\sin ^2\left(x\right)}\sin ^2\left(x\right)$

$\mathrm{Multiply\:fractions}:\quad \:a\cdot \dfrac{b}{c}=\dfrac{a\:\cdot \:b}{c}$

$=\dfrac{\cos ^2\left(x\right)\sin ^2\left(x\right)}{\sin ^2\left(x\right)}$

$\mathrm{Cancel\:the\:common\:factor:}\:\sin ^2\left(x\right)$

$=\cos ^2\left(x\right)$

$\left(\dfrac{\sin \left(x\right)}{\cos \left(x\right)}\right)^2\cos ^2\left(x\right)$

$\left(\dfrac{\sin \left(x\right)}{\cos \left(x\right)}\right)^2$

$\mathrm{Apply\:exponent\:rule}:\quad \left(\dfrac{a}{b}\right)^c=\dfrac{a^c}{b^c}$

$=\dfrac{\sin ^2\left(x\right)}{\cos ^2\left(x\right)}$

$=\dfrac{\sin ^2\left(x\right)}{\cos ^2\left(x\right)}\cos ^2\left(x\right)$

$\mathrm{Multiply\:fractions}:\quad \:a\cdot \dfrac{b}{c}=\dfrac{a\:\cdot \:b}{c}$

$=\dfrac{\sin ^2\left(x\right)\cos ^2\left(x\right)}{\cos ^2\left(x\right)}$

$\mathrm{Cancel\:the\:common\:factor:}\:\cos ^2\left(x\right)$

$=\sin ^2\left(x\right)$

$=\cos ^2\left(x\right)+\sin ^2\left(x\right)$

$\mathrm{Use\:the\:following\:identity}:\quad \cos ^2\left(x\right)+\sin ^2\left(x\right)=1$

$=1$