Question

If tan2 θ = 1 − k2, show that sec θ + tan3 θ cosec θ = (2−k2)3/2. Also, find the values of k for which this result holds.
​

Answer 1

${\green{\boxed{\boxed{\underline{\purple{\red{\rm{QUESTION :}}}}}}}}$

If tan² θ = 1 − k², show that sec θ + tan³θ cosec θ = (2−k²)3/2. Also, find the values of k for which this result holds.

${\green{\boxed{\boxed{\underline{\purple{\red{\rm{SOLUTION :}}}}}}}}$

$\leadsto \rm \: sec \theta \: + tan {}^{3} \theta \: \: cosec \theta \\ \\ \leadsto \rm \: ( \frac{1}{cosec \theta}) + (\frac{sin \theta}{cos \theta} ) {}^{3} \: \: ( \frac{1}{sin \theta}) \\ \\ \leadsto \rm \: (\frac{1}{cos \theta} ) + ( \frac{ sin {}^{3} \theta }{cos {}^{3} \theta} ) \: \: ( \frac{1}{sin \theta} ) \\ \\ \leadsto \rm \: ( \frac{1}{cos \theta} ) + ( \frac{sin {}^{2} \theta}{cos {}^{3} \theta} ) \: \\ \\ \: \leadsto \rm \: ( \frac{cos {}^{2} \theta + sin {}^{2}\theta }{cos {}^{3}\theta } ) \\ \\ \leadsto \rm \: \frac{1}{cos {}^{3} \theta} \\ \\ \leadsto \rm \: sec {}^{3} \theta \\ \\ \leadsto \rm \: \red{ (sec {}^{2} \theta) \: sec \theta \: -------(1) }\: \\ \\ \leadsto \green{ \rm{ \underline{Given That :}}} \\ \\ \: \leadsto \rm \: tan {}^{2} \theta \: = 1 - k {}^{2} \: \: \\ \\ \leadsto \green{ \rm{ \underline{ add \: 1 \: on \: both \: sides : }}} \: \\ \\ \leadsto \rm \: 1 + tan {}^{2} \theta = 1 + 1 - k {}^{2} \\ \\ \leadsto \green{ \rm{ \underline{ by \: applying \: the \: above \: values \: in \: }}} \: \: \\ \green{ \rm{ \underline{ the \: first \: equation \red{ \: we \: get : }}}} \\ \\ \leadsto \rm\: (2 - k {}^{2}) \: \: \sqrt{(2 - k {}^{2} )} \\ \\ \leadsto \rm\: \: (2 - k {}^{2} ) \frac{3}{2} \\ \\ \leadsto \rm\: \: { \boxed{ \boxed{ \underline { \rm{ \pink{The \: value \: of \: K \: is \: \: [ -1 , 1 ] \: }}}}}}$

Answer 2

k = 1 (2((#($8383829299292(3($($($($