MCQ
$\frac{\tan ^2 \theta}{1+\tan ^2 \theta}+\frac{\cot ^2 \theta}{1+\cot ^2 \theta}=$
- ✓1
- B$2 \tan ^2 \theta$
- C$2 \cot ^2 \theta$
- D$2 \sec ^2 \theta$
✓
Answer
Correct option: A.
1
(A) 1
$\frac{\tan ^2 \theta}{1+\tan ^2 \theta}+\frac{\cot ^2 \theta}{1+\cot ^2 \theta}$
$=\frac{\tan ^2 \theta}{\sec ^2 \theta}+\frac{\cot ^2 \theta}{\operatorname{cosec}^2 \theta}$
$=\tan ^2 \theta \times \frac{1}{\sec ^2 \theta}+\cot ^2 \theta \times \frac{1}{\operatorname{cosec}^2 \theta}$
$=\frac{\sin ^2 \theta}{\cos ^2 \theta} \times \cos ^2 \theta+\frac{\cos ^2 \theta}{\sin ^2 \theta} \times \sin ^2 \theta$
$=\sin ^2 \theta+\cos ^2 \theta=1$
$\frac{\tan ^2 \theta}{1+\tan ^2 \theta}+\frac{\cot ^2 \theta}{1+\cot ^2 \theta}$
$=\frac{\tan ^2 \theta}{\sec ^2 \theta}+\frac{\cot ^2 \theta}{\operatorname{cosec}^2 \theta}$
$=\tan ^2 \theta \times \frac{1}{\sec ^2 \theta}+\cot ^2 \theta \times \frac{1}{\operatorname{cosec}^2 \theta}$
$=\frac{\sin ^2 \theta}{\cos ^2 \theta} \times \cos ^2 \theta+\frac{\cos ^2 \theta}{\sin ^2 \theta} \times \sin ^2 \theta$
$=\sin ^2 \theta+\cos ^2 \theta=1$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Mean of a certain number of observation is. If each observation is divided by $m(m \neq 0)$ and increased by $n,$ then the mean of new observation is :
→A solid frustum is of height $8\ cm$. If the radii of its lower and upper ends are $3\ cm$ and $9\ cm$ respectively, then its slant height is
→The height of a cone is $30\ cm$. A small cone is cut off at the top by a plane parallel to the base. If its volume be of the volume of the given cone, then the height above the base at which the section has been made, is:
→In figure, $\text{QR}$ is a common tangent to the given circles, touching externally at the point $T$. The tangent at $T$ meets $\text{QR}$ at $P$. If $\text{QP} =3.8$, then the length of $\text{QR} ($in $cm )$ is :
→A piece of paper in the shape of a sector of a circle (see figure 1) is rolled up to form a right-circular cone (see figure 2). The value of angle $\theta$ is:
→$\triangle ABC \sim \triangle DEF$ such that $\operatorname{ar}(\triangle ABC )=64 cm^2$ and $\operatorname{ar}(\triangle DEF )=81 cm^2$. Then, the ratio of their corresponding sides is
→Value of the middle $-$ most observation $(s)$ is called :
→Given that $\sin \theta=\frac{a}{b}$, then $\tan \theta$ is equal to
→Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion : $ (\cos^4\text{A}-\sin^4\text{A})$ is equal to $ 2\cos^2\text{A}-1.$
Reason : The value of $\cos\theta$ decreases as $\theta$ increases.
→Assertion : $ (\cos^4\text{A}-\sin^4\text{A})$ is equal to $ 2\cos^2\text{A}-1.$
Reason : The value of $\cos\theta$ decreases as $\theta$ increases.
$\tan ^4 \theta+\tan ^2 \theta=$
→