Question

(cosec² theta -1) tan²theta =1​

Answer 1

Question:-

$\implies\rm ( \csc {}^{2} \theta - 1) \tan {}^{2} ( \theta) = 1$

Solution:-

$\implies \bigg(\dfrac{1}{ { \sin }^{2} \theta} - 1 \bigg) \tan {}^{2} ( \theta) = 1$

$\rm \implies \bigg( \dfrac{1 - \sin {}^{2} \theta }{ \sin {}^{2} \theta} \bigg) \tan {}^{2} \theta = 1$

$\implies \bigg( \dfrac{ \cos {}^{2} \theta }{ \sin {}^{2} \theta } \bigg) \tan {}^{2} \theta = 1$

$\rm \implies \dfrac{ \cos {}^{2} \theta }{ \sin {}^{2} \theta} \times \dfrac{ \sin {}^{2} \theta}{ \cos {}^{2} \theta}$

$\rm \implies \dfrac{ \cancel{\cos {}^{2} \theta }}{ \cancel{\sin {}^{2} \theta}} \times \dfrac{ \cancel{\sin {}^{2} \theta}}{ \cancel{ \cos {}^{2} \theta}} = 1$

$\implies1 = 1$

Hence proved

Some trigonometry identity

$\to \tan( \theta) = \dfrac{ \sin( \theta) }{ \cos( \theta) }$

$\to \cot( \theta) = \dfrac{ \cos( \theta) }{ \sin( \theta) }$

$\to \sec( \theta) = \dfrac{1}{ \cos( \theta) }$

$\to \: \csc( \theta) = \dfrac{1}{ \sin( \theta) }$

$\to \: \sin {}^{2} ( \theta) + \cos {}^{2} ( \theta) = 1$

Answer 2

Answer:

$Question:-</p><p></p><p>\implies\rm ( \csc {}^{2} \theta - 1) \tan {}^{2} ( \theta) = 1⟹(csc2θ−1)tan2(θ)=1</p><p> </p><p>Solution:-</p><p></p><p>\implies \bigg(\dfrac{1}{ { \sin }^{2} \theta} - 1 \bigg) \tan {}^{2} ( \theta) = 1⟹(sin2θ1−1)tan2(θ)=1</p><p></p><p>\rm \implies \bigg( \dfrac{1 - \sin {}^{2} \theta }{ \sin {}^{2} \theta} \bigg) \tan {}^{2} \theta = 1⟹(sin2θ1−sin2θ)tan2θ=1</p><p></p><p>\implies \bigg( \dfrac{ \cos {}^{2} \theta }{ \sin {}^{2} \theta } \bigg) \tan {}^{2} \theta = 1⟹(sin2θcos2θ)tan2θ=1</p><p></p><p>\rm \implies \dfrac{ \cos {}^{2} \theta }{ \sin {}^{2} \theta} \times \dfrac{ \sin {}^{2} \theta}{ \cos {}^{2} \theta}⟹sin2θcos2θ×cos2θsin2θ</p><p></p><p>\rm \implies \dfrac{ \cancel{\cos {}^{2} \theta }}{ \cancel{\sin {}^{2} \theta}} \times \dfrac{ \cancel{\sin {}^{2} \theta}}{ \cancel{ \cos {}^{2} \theta}} = 1⟹sin2θcos2θ×cos2θsin2θ=1</p><p></p><p>\implies1 = 1⟹1=1</p><p></p><p>Hence proved</p><p></p><p>Some trigonometry identity</p><p></p><p>\to \tan( \theta) = \dfrac{ \sin( \theta) }{ \cos( \theta) }→tan(θ)=cos(θ)sin(θ)</p><p></p><p>\to \cot( \theta) = \dfrac{ \cos( \theta) }{ \sin( \theta) }→cot(θ)=sin(θ)cos(θ)</p><p></p><p>\to \sec( \theta) = \dfrac{1}{ \cos( \theta) }→sec(θ)=cos(θ)1</p><p></p><p>\to \: \csc( \theta) = \dfrac{1}{ \sin( \theta) }→csc(θ)=sin(θ)1</p><p></p><p>\to \: \sin {}^{2} ( \theta) + \cos {}^{2} ( \theta) = 1→sin2(θ)+cos2(θ)=1</p><p></p><p>$