Question

Hey please explain in it I stuck there please solve fast

If ( sin x + cosec x ) ² + ( cos x + sec x )² = k + tan² x + cot² x then what is the value of k.
a- 8
b - 7
c- 4
d-3
Don't Spam ​

Answer 1

`\huge{ \bold{Answer -: }}`

Option b is correct

Explanation-:

`\large \bf{given}`

`\large{ \sf{ \: (sin \: x \: + cosec \: x) {}^(2) + (cos \: x + sec \: x) {}^(2) }}`

`\large{ \sf{ = k + { \tan^(2) x + \cot^(2) x}}}`

`\large{ \sf{\implies \: sin {}^(2) x + cosec {}^(2) x + 2 + {cos}^(2) x + {sec}^(2) x + 2}}`

`\large{ \sf{ = k + {tan}^(2) x + {cot}^(2) x}}`

`\large{ \sf{ \implies \: 1 + {cosec}^(2) x - {cot}^(2) x + {sec}^(2) x - {tan}^(2) x + 4 = k}}`

`\large{ \sf{ \pink{ \implies \: 1 + 1 + 1 + 4 = k \implies \: k = 7}}}`

`\therefore \: k \: = 7`

____________________

Answer 2

b) 7

Explanation:

Given equation:

`(\sin x + \csc x)^2+(\cos x + \sec x)^2=k+\tan^2 x + \cot^2 x`

Expand the left side of the equation:

`\implies \sin^2x+2 \sin x \csc x + \csc^2x + \cos^2 x + 2 \cos x \sec x + \sec^2 x`

Simplify:

`\implies \sin^2x+2 \sin x \cdot (1)/(\sin x) + \csc^2x + \cos^2 x + 2 \cos x \cdot (1)/(\cos x) + \sec^2 x`

`\implies \sin^2x+2 + \csc^2x + \cos^2 x + 2 + \sec^2 x`

`\implies \sec^2 x+\csc^2x+ \sin^2x + \cos^2 x +4`

Use the identity: sin²x + cos²x = 1

`\implies \sec^2 x+ \csc^2x+1+4`

`\implies \sec^2 x+ \csc^2x+5`

Use the identities: sec²x = tan²x + 1 and csc²x = cot²x + 1

`\implies \tan^2x +1 +\cot^2x +1 +5`

`\implies 7+\tan^2x + \cot^2x`

Therefore:

`(\sin x + \csc x)^2+(\cos x + \sec x)^2=7+\tan^2 x + \cot^2 x`