Tag: trigonometrical ratio and identities

Questions Related to trigonometrical ratio and identities

$\sin ^{ 8 }{ \theta  } -\cos ^{ 8 }{ \theta  } -\left( \sin ^{ 2 }{ \theta  } -\cos ^{ 2 }{ \theta  }  \right) \left( 1-\sin ^{ 2 }{ \theta  }  \right) $=0

maths trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities trigonometrical ratio and identities

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

$\sin^4 \theta+\cos^4 \theta=(\sin^2 \theta+\cos^2 \theta)^2-2\sin^2 \theta\cos^2 \theta=1-2\sin^2 \theta\cos^2 \theta$

$\sin^4 \theta-\cos^4 \theta=(\sin^2 \theta-\cos^2 \theta)(\sin^2 \theta+\cos^2 \theta)=\sin^2 \theta-\cos^2 \theta$
$\sin^8\theta-\cos^8\theta-(\sin^2 \theta-\cos^2 \theta)(1-\sin^2 \theta)=(\sin^4 \theta+\cos^4 \theta)(\sin^4 \theta-\cos^4 \theta)-(\sin^2 \theta-\cos^2 \theta)(1-\sin^2 \theta)$
                                                                                 $=(1-2\sin^2 \theta\cos^2 \theta)(\sin^2 \theta-\cos^2 \theta)-(\sin ^2 \theta-\cos^2\theta)(1-\sin^2 \theta)$
                                                                                 $=(\sin^2 \theta-\cos^2 \theta)(\sin^2 \theta)(1-2\cos^2 \theta)$
So the given relation is $\text{False}$

If $tan x + cot x = 2$, then $sin^{2n}x+cos^{2n}x=$

maths compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric ratios of acute angles trigonometric identities trigonometrical ratio and identities

  1. $\dfrac{1}{2}$

  2. $2^n$

  3. $\dfrac{1}{2^n}$

  4. $\dfrac{1}{2^{n-1}}$

Show answer
Correct Option: D
Explanation:

Given $\tan x+\cot x=2$

$\implies \tan x+\dfrac{1}{\tan x}=2$
$\implies \tan^2 x-2\tan x+1=0$
$\implies (\tan x-1)^2=0$
$\implies \tan x=1\implies x=\dfrac{\pi}{4}$
$\sin^{2 n} x+\cos^{2 n} x=\bigg(\dfrac{1}{\sqrt{2}}\bigg)^{2 n}+\bigg(\dfrac{1}{\sqrt{2}}\bigg)^{2 n}=\dfrac{1}{2^n}+\dfrac{1}{2^n}=\dfrac{2}{2^n}=\dfrac{1}{2^{n-1}}$