Question

Consider the following equations:
1. $co{\sec }^{2}x+{\sec }^{2}x=co{\sec }^{2}x{\sec }^{2}x$
2. ${\sec }^{2}x+{\tan }^{2}x={\sec }^{2}x{\tan }^{2}x$
3. $co{\sec }^{2}x+{\tan }^{2}x={\cot }^{2}x+{\sec }^{2}x$
Which of the above statements are correct?

• A. $1$ and $2$ only
• B. $2$ and $3$ only
• C. $1$ and $3$ only
• D. $1,2$ and $3$

Answer 1

The correct option is C $1$ and $3$ only

$\bullet$ $cose{c}^{2}x+se{c}^{2}x=\frac{1}{si{n}^{2}x}+\frac{1}{co{s}^{2}x}=\frac{si{n}^{2}x+co{s}^{2}x}{si{n}^{2}x+co{s}^{2}x}$

$=cose{c}^{2}xse{c}^{2}x$

$\bullet$ $se{c}^{2}x+ta{n}^{2}x=\frac{1+si{n}^{2}x}{co{s}^{2}x}\ne \frac{si{n}^{2}x}{co{s}^{4}x}$

$\bullet$ $cose{c}^{2}x+ta{n}^{2}x=co{t}^{2}x+se{c}^{2}x$

$\Rightarrow cose{c}^{2}x-co{t}^{2}x=se{c}^{2}x-ta{n}^{2}x=1$

(By trigo identities)