Question
Very-Short and Short-Answer Questions:
If $\sec\theta=\text{x},$ write the value of $\tan\theta.$
If $\sec\theta=\text{x},$ write the value of $\tan\theta.$
✓
Answer
$\sec\theta=\text{x}$
$\Rightarrow\frac{1}{\cos\theta}=\text{x}$
$\Rightarrow\cos\theta=\frac{1}{\text{x}}$
$\Rightarrow\cos^2\theta=\frac{1}{\text{x}^2}$
$\Rightarrow\sin^2\theta=1-\cos^2\theta=1-\frac{1}{\text{x}^2}=\frac{\text{x}^2-1}{\text{x}}$
$\Rightarrow\sin\theta=\frac{\sqrt{\text{x}^2-1}}{\text{x}}$
Now, $\tan\theta=\frac{\sin\theta}{\cos\theta}$
$=\frac{\frac{\sqrt{\text{x}^2-1}}{\text{x}}}{\frac{1}{\text{x}}}=\sqrt{\text{x}^2-1}$
$\Rightarrow\frac{1}{\cos\theta}=\text{x}$
$\Rightarrow\cos\theta=\frac{1}{\text{x}}$
$\Rightarrow\cos^2\theta=\frac{1}{\text{x}^2}$
$\Rightarrow\sin^2\theta=1-\cos^2\theta=1-\frac{1}{\text{x}^2}=\frac{\text{x}^2-1}{\text{x}}$
$\Rightarrow\sin\theta=\frac{\sqrt{\text{x}^2-1}}{\text{x}}$
Now, $\tan\theta=\frac{\sin\theta}{\cos\theta}$
$=\frac{\frac{\sqrt{\text{x}^2-1}}{\text{x}}}{\frac{1}{\text{x}}}=\sqrt{\text{x}^2-1}$
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
A well of diameter 3m is dug 14m deep. The earth taken out of it is spread evenly all around it to a width of 4m to form an embankment. Find the height of the embankment.
In the following, determine whether the given quadratic equation have real root and if so, find the root:
$3x^2 - 5x + 2 = 0$
→$3x^2 - 5x + 2 = 0$
Solve the following quadratic equations by factorisation: 3x² - x - 10 = 0
Solve for x.
$\frac{1}{\text{x}-3}-\frac{1}{\text{x}+5}=\frac{1}{6},$ $\text{x}\neq3,-5$
→$\frac{1}{\text{x}-3}-\frac{1}{\text{x}+5}=\frac{1}{6},$ $\text{x}\neq3,-5$
For the following arithmetic progressions write the first term a and the common difference d:
$\frac{1}{5},\frac{3}{5},\frac{5}{5},\frac{7}{5}.$
→$\frac{1}{5},\frac{3}{5},\frac{5}{5},\frac{7}{5}.$
If Three coins are tossed simultaneously, find the probability of the
following events.
i) Event A : To get no heads.
ii) Event B : To get at least two heads.
following events.
i) Event A : To get no heads.
ii) Event B : To get at least two heads.
Solve the following simultaneous equations using Cramer’s rule.
x + 2y = –1; 2x – 3y = 12
x + 2y = –1; 2x – 3y = 12
D and E are points on the sides AB and AC respectively of a $\triangle\text{ABC}$ such that DE || BC: If $\frac{\text{AD}}{\text{AB}}=\frac{8}{15}$ and EC = 3.5cm, find AE.
→In a right triangle $ABC,$ right-angled at $B, D$ is a point on hypotenuse such that $\text{BD}\perp\text{AC}.$ If $\text{DP}\perp\text{AB}$ and $\text{DQ}\perp\text{BC}$ then prove that.
→- $\text{DQ}^2=\text{DP}.\text{DQ}$
- $\text{DP}^2=\text{DQ}.\text{AP}$
If the radii of the circular ends of a conical bucket which is $45\ cm$ high be $28\ cm$ and $7\ cm$, find the capacity of the bucket. $(\text{use}\ \pi=\frac{22}{7}).$
→