Question

If $(\sin \alpha +\cos ec\alpha {)}^2+(\cos \alpha +\sec \alpha {)}^2=K+{\tan }^2\alpha +{\cot }^2\alpha$, then $K=$____

• A. $9$
• B. $7$
• C. $5$
• D. $3$

Answer 1

The correct option is B $7$

Given $(\sin \alpha +\csc \alpha {)}^2+(\cos \alpha +\sec \alpha {)}^2=K+{\tan }^2\alpha +{\cot }^2\alpha$

$\Rightarrow {\sin }^2\alpha +{\csc }^2\alpha +2\sin \alpha \csc \alpha +{\cos }^2\alpha +{\sec }^2\alpha +2\cos \alpha \sec \alpha )=K+{\tan }^2\alpha +{\cot }^2\alpha$

we know, that $\sin \alpha \csc \alpha =1$ and $\cos \alpha \sec \alpha =1$

And also Trignometric identity: ${\sin }^2\alpha +{\cos }^2\alpha =1$

$\therefore 1+{\csc }^2\alpha +2+{\sec }^2\alpha +2-{\tan }^2\alpha -{\cot }^2\alpha =K$

$\Rightarrow 5+{\csc }^2\alpha -{\cot }^2\alpha +{\sec }^2\alpha -{\tan }^2\alpha =K$

Trignometric identities: ${\csc }^2\alpha -{\cot }^2\alpha =1$ and ${\sec }^2\alpha -{\tan }^2\alpha =1$

$\Rightarrow 5+1+1=K$

$\Rightarrow K=7$