Answer:
b) 7
Step-by-step explanation:
Given equation:
[tex](\sin x + \csc x)^2+(\cos x + \sec x)^2=k+\tan^2 x + \cot^2 x[/tex]
Expand the left side of the equation:
[tex]\implies \sin^2x+2 \sin x \csc x + \csc^2x + \cos^2 x + 2 \cos x \sec x + \sec^2 x[/tex]
Simplify:
[tex]\implies \sin^2x+2 \sin x \cdot \dfrac{1}{\sin x} + \csc^2x + \cos^2 x + 2 \cos x \cdot \dfrac{1}{\cos x} + \sec^2 x[/tex]
[tex]\implies \sin^2x+2 + \csc^2x + \cos^2 x + 2 + \sec^2 x[/tex]
[tex]\implies \sec^2 x+\csc^2x+ \sin^2x + \cos^2 x +4[/tex]
Use the identity: sin²x + cos²x = 1
[tex]\implies \sec^2 x+ \csc^2x+1+4[/tex]
[tex]\implies \sec^2 x+ \csc^2x+5[/tex]
Use the identities: sec²x = tan²x + 1 and csc²x = cot²x + 1
[tex]\implies \tan^2x +1 +\cot^2x +1 +5[/tex]
[tex]\implies 7+\tan^2x + \cot^2x[/tex]
Therefore:
[tex](\sin x + \csc x)^2+(\cos x + \sec x)^2=7+\tan^2 x + \cot^2 x[/tex]
[tex] \huge{ \bold{Answer -: }}[/tex]
Option b is correct
[tex] \large \bf{given}[/tex]
[tex] \large{ \sf{ \: (sin \: x \: + cosec \: x) {}^{2} + (cos \: x + sec \: x) {}^{2} }}[/tex]
[tex]\large{ \sf{ = k + { \tan^{2} x + \cot^{2} x}}}[/tex]
[tex] \large{ \sf{\implies \: sin {}^{2} x + cosec {}^{2} x + 2 + {cos}^{2} x + {sec}^{2} x + 2}}[/tex]
[tex]\large{ \sf{ = k + {tan}^{2} x + {cot}^{2} x}}[/tex]
[tex]\large{ \sf{ \implies \: 1 + {cosec}^{2} x - {cot}^{2} x + {sec}^{2} x - {tan}^{2} x + 4 = k}}[/tex]
[tex]\large{ \sf{ \pink{ \implies \: 1 + 1 + 1 + 4 = k \implies \: k = 7}}}[/tex]
[tex] \therefore \: k \: = 7[/tex]
