Question

Which of the following statements is/are correct?

1. cos2A = $\frac{1+ta{n}^{2}A}{1-ta{n}^{2}A}$

2. $cose{c}^{2}A$ = 1 + $co{t}^2\theta$

3. $se{c}^2\theta$ = 1 + $co{s}^2\theta$

• A. Only 1
• B. Only 2
• C. Only 1 & 2
• D. Only 2 & 3

Answer 1

The correct option is B

Only 2

We know,

cos2A = $co{s}^{2}A-si{n}^{2}A$

In statement 1, cos2A is given in terms of tanA. Multiple & divide the RHS by $co{s}^{2}A$

cos2A = $\frac{(co{s}^{2}A-si{n}^{2}A)\times co{s}^{2}A}{co{s}^{2}A}$

= $(1-ta{n}^{2}A)co{s}^{2}A$

= $\frac{1-ta{n}^{2}A}{se{c}^{2}A}$ = $\frac{1-ta{n}^{2}A}{1+ta{n}^{2}A}$ {$se{c}^{2}A=1+ta{n}^{2}A$}

Statement 1 is incorrect.

Statement 2 is an identity.

$cose{c}^2\theta$ = 1 + $co{t}^2\theta$

We know , $si{n}^2\theta$ + $co{s}^2\theta$ = 1

Divide both side by $si{n}^2\theta$

$\frac{si{n}^2\theta +co{s}^2\theta }{si{n}^2\theta }$ = $\frac{1}{si{n}^2\theta }$

1 + $co{t}^2\theta$ = $cose{c}^2\theta$

Statement 2 is correct.

We know, $se{c}^2\theta$ = $\frac{1}{co{s}^2\theta }$.But statement 3 is given $se{c}^2\theta$ = 1 + $co{s}^2\theta$.So,statement 3 is incorrect.

$se{c}^2\theta$ = 1 + $ta{n}^2\theta$

We know,$si{n}^2\theta$ + $co{s}^2\theta$ = 1

Divide both the sides of identity by$co{s}^2\theta$

$\frac{si{n}^2\theta +co{s}^2\theta }{co{s}^2\theta }$ = $\frac{1}{co{s}^2\theta }$

$ta{n}^2\theta +1=se{c}^2\theta$