Question

If $A$ and $B$ are complementary angles, then

• A. $\sin \left(A\right)=\sin \left(B\right)$
• B. $\cos \left(A\right)=\cos \left(B\right)$
• C. $\tan \left(A\right)=\tan \left(B\right)$
• D. $\sec \left(A\right)=\mathrm{cosec}\left(B\right)$

Answer 1

The correct option is D

$\sec \left(A\right)=\mathrm{cosec}\left(B\right)$

Given, $A$ and $B$ are complementary angles.

$\Rightarrow$ $m\angle A+m\angle B={90}^{\circ }$

$\Rightarrow$ $A=90-B$ $...\left(i\right)$

We know that ,

$\sin \theta =\cos \left(90-\theta \right)$

Taking reciprocal on both sides we get,

$\Rightarrow$ $\frac{1}{\sin \theta }=\frac{1}{\cos \left(90-\theta \right)}$

$\Rightarrow$ $\cos ec\theta =sec\left(90-\theta \right)$

Now, replace $\theta$ by $B$

$\Rightarrow$ $\cos ecB=sec\left(90-B\right)$

$\Rightarrow$ $\cos ecB=secA$ [From $\left(i\right)$]

Thus, $\sec \left(A\right)=\mathrm{cosec}\left(B\right)$,

Hence, option D is correct.