Question
If $\sec A = \sqrt{2} ,$ find $:\frac{3 \cot ^2 A +2 \sin ^2 A }{\tan ^2 A -\cos ^2 A }.$
✓
Answer
$\sec A=\frac{\sqrt{2}}{1}$
i.e. $\frac{\text { hypotenuse }}{\text { base }}=\frac{ AC }{ AB }=\frac{\sqrt{2}}{1}$
Therefore if length of base $= x ,$ length of hypotenuse $=\sqrt{2} x$
Since
$AB^2 + BC^2 = AC^2\dots ...[$Using Pythagoras Theorem$]$
$(\sqrt{2} x)^2-(x)^2= BC ^2$
$BC ^2=2 x^2-x^2$
$BC ^2= x ^2$
$\therefore BC = x$
Now
$\cos A =\frac{1}{\sec A }=\frac{1}{\sqrt{2}}$
$\sin A =\frac{ BC }{ AC }=\frac{1}{\sqrt{2}}$
$\tan A =\frac{ BC }{ AB }=1$
$\cot A=\frac{1}{\tan A}=1$
Therefore
$\frac{3 \cot ^2 A +2 \sin ^2 A }{\tan ^2 A -\cos ^2 A }=\frac{3(1)^2+2\left(\frac{1}{\sqrt{2}}\right)^2}{1^2-\left(\frac{1}{\sqrt{2}}\right)^2}$
$=\frac{3+1}{1-\frac{1}{2}}$
$=\frac{4}{\frac{1}{2}}$
$=4 \times \frac{2}{1}$
$=8$
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
More "[4 marks sum]" from Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals]→Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals] — all question types→MATHEMATICS STD 9 question papers→ICSE Board STD 9 question papers→ICSE Board question paper generator→
Similar questions
If $\left(x+\frac{1}{x}\right)=5$, find the values of $\left(x^2+\frac{1}{x^2}\right)$
→The cost of fencing a circular field at the rate of $Rs. 240$ per meter is $Rs.52,800$. The field is to be ploughed at the rate of $Rs. 12.50$ per $\sim m^2$. Find the cost of pouching the field.
→A boy played $100$ games, gaining $Rs. 50 $ on each game that he won, and losing $Rs. 20$ for each game that he lost. If on the whole he gained $Rs. 800$, find the number of games that he won.
→
In the given figure, ABCD is a rectangle inscribed in a circle. If two adjacent sides of the rectangle be 8 cm and 6 cm, calculate :
(i) the radius of the circle; and
(ii) the area of the shaded region. (Take $\pi$ = 3.14 ).
$\left[\right.$ Hint. Radius $=\left(\frac{1}{2} \times\right.$ Diagonal $\left.)=5 cm\right]$
Find the altitude of an equilateral triangle of side $5 \sqrt{3} cm$.
→A sum of money is lent at $8\%$ per annum compound interest. If the interest for the second year exceeds that for the first year by $Rs. 96$, find the sum of money.
→A farmer has an increase of 12.5% in the output of wheat in his farm every year. This year, he produced 2916 quintals of wheat. What was his annual production of wheat 2 years ago?
A man borrows $Rs. 10,000$ at $5\%$ per annum compound interest. He repays $35\%$ of the sum borrowed at the end of the first year and $42\%$ of the sum borrowed at the end of the second year. How much must he pay at the end of the third year in order to clear the debt ?
→Find the scale factor in each of the following and state the type of size transformation:Actual area $= 64m^2$, Model area $= 100\ cm^2$
→Solve the following pair of linear $($Simultaneous $)$ equation using method of elimination by substitution $:2( x - 3 ) + 3( y - 5 ) = 0,5( x - 1 ) + 4( y - 4 ) = 0$
→