Question
Prove that $(cosec A - \sin A)( \sec A - \cos A) \sec^2 A = tan A$.
✓
Answer
LHS = $(cosec A - \sin A)(\sec A - \cos A). \sec^2A$
$=\left(\frac{1}{\sin A}-\sin A\right) \cdot\left(\frac{1}{\cos A}-\cos A\right) \cdot \frac{1}{\cos ^2 A}$
$=\frac{1-\sin ^2 A}{\sin A} \cdot \frac{1-\cos ^2 A}{\cos A} \times \frac{1}{\cos ^2 A}$
$=\frac{\cos ^2 A}{\sin A} \times \frac{\sin ^2 A}{\cos A} \times \frac{1}{\cos ^2 A} \ldots\left[\because\left(1-\sin ^2 A\right)=\cos ^2 A, 1-\cos ^2 A=\sin ^2 A\right]$
$ =\frac{\sin A}{\cos A}=\tan A$
$= RHS $
Hence proved.
$=\left(\frac{1}{\sin A}-\sin A\right) \cdot\left(\frac{1}{\cos A}-\cos A\right) \cdot \frac{1}{\cos ^2 A}$
$=\frac{1-\sin ^2 A}{\sin A} \cdot \frac{1-\cos ^2 A}{\cos A} \times \frac{1}{\cos ^2 A}$
$=\frac{\cos ^2 A}{\sin A} \times \frac{\sin ^2 A}{\cos A} \times \frac{1}{\cos ^2 A} \ldots\left[\because\left(1-\sin ^2 A\right)=\cos ^2 A, 1-\cos ^2 A=\sin ^2 A\right]$
$ =\frac{\sin A}{\cos A}=\tan A$
$= RHS $
Hence proved.
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
The sum of 2nd and 7th terms of an A.P. is 30. If its 15th term is 1 less than twice its 8th term, find the A.P.
Find three consecutive odd integers, the sum of whose squares is $83$.
→i) In the given figure 3 × CP = PD = 9cm and AP = 4.5 cm Find BP.
The total area of a solid metallic sphere is $1256 \ cm^2.$ It is melted and recast into solid right circular cones of radius $2.5 \ cm$ and height $8 \ cm.$ Calculate: the radius of the solid sphere.
→Solve the following by reducing them to quadratic equations:
$x^4 - 26x2 + 25 = 0$
→$x^4 - 26x2 + 25 = 0$
A horse is tied with a 21 m laig rope to the corner of a field which is in the shape of an equilateral triangle. Find the area of the field over which it can graze.
If $\frac{a}{b}=\frac{c}{d}=\frac{r}{f}$, prove that $\left(\frac{a^2 b^2+c^2 d^2+e^2 f^2}{a b^3+c d^3+e f^3}\right)^{\frac{3}{2}}=\sqrt{\frac{a c e}{b d f}}$
→From the figure , given below , calculate the lenth of CD .
Solve the following equation:
$\frac{\cos ^2 \theta-3 \cos \theta+2}{\sin ^2 \theta}=1$.
→$\frac{\cos ^2 \theta-3 \cos \theta+2}{\sin ^2 \theta}=1$.
Zafarullah has a recurring deposit The list price of the television be? $12,500.$ account in a bank for $3^{1/2}$ years at $9.5\%$ S.I. p.a. If he gets $Rs. 78,638$ at the time of maturity. Find the monthly instalment.
→