Question
Prove the identity $(\sin \theta + \cos \theta) (\tan \theta + \cot \theta ) = \sec \theta + \cos ec θ.$
✓
Answer
$\text { L.H.S. }=(\sin \theta+\cos \theta)(\tan \theta+\cot \theta)$
$=(\sin \theta+\cos \theta)\left(\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right)$
$=(\sin \theta+\cos \theta)\left(\frac{\sin ^2 \theta+\cos ^2 \theta}{\cos \theta \sin \theta}\right)$
$=(\sin \theta+\cos \theta) \times \frac{1}{\sin \theta \cos \theta}\left(\because \sin ^2 \theta+\cos ^2 \theta=1\right)$
$=\frac{\sin \theta+\cos \theta}{\cos \theta \sin \theta}$
$=\frac{\sin \theta}{\cos \theta \sin \theta}+\frac{\cos \theta}{\cos \theta \sin \theta}$
$=\frac{1}{\cos \theta}+\frac{1}{\sin \theta}$
$=\sec \theta+\cos e c \theta$
$= R.H.S$
Hence proved.
$=(\sin \theta+\cos \theta)\left(\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right)$
$=(\sin \theta+\cos \theta)\left(\frac{\sin ^2 \theta+\cos ^2 \theta}{\cos \theta \sin \theta}\right)$
$=(\sin \theta+\cos \theta) \times \frac{1}{\sin \theta \cos \theta}\left(\because \sin ^2 \theta+\cos ^2 \theta=1\right)$
$=\frac{\sin \theta+\cos \theta}{\cos \theta \sin \theta}$
$=\frac{\sin \theta}{\cos \theta \sin \theta}+\frac{\cos \theta}{\cos \theta \sin \theta}$
$=\frac{1}{\cos \theta}+\frac{1}{\sin \theta}$
$=\sec \theta+\cos e c \theta$
$= R.H.S$
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
A bucket is raised from a well by means of a rope which is wound round a wheel of diameter $77\ cm.$ Given that the ascends in $1$ minute $28$ seconds with a uniform speed of $1.1\ m/sec,$ calculate the number of complete revolutions the wheel makes in raising the bucket. (Take $\pi =22/7$)
→Solve the following equation by factorization$4 \sqrt{3} x^2+5 x-2 \sqrt{3}=0$
→A (1, 1), B (5, 1), C (4, 2) and D (2, 2) are vertices of a quadrilateral. Name the quadrilateral ABCD. A, B, C, and D are reflected in the origin on to A’, B’, C’ and D’ respectively. Locate A’, B’, C’ and D’ on the graph sheet and write their co-ordinates .
Are D, A, A’ and D’ collinear?
Find the $\%$ return on investing in :
(i) $6\%$ ₹ $ 100$ shares at ₹ $ 120$.$\quad$(ii) $8 \frac{1}{3} \%$ ₹ $100$ shares at ₹ $150.$
→(i) $6\%$ ₹ $ 100$ shares at ₹ $ 120$.$\quad$(ii) $8 \frac{1}{3} \%$ ₹ $100$ shares at ₹ $150.$
Ameesha loaned $Rs. 24,000$ to a friend for $2 \frac{1}{2}$ at $10 \%$ p.a. compound interest.
Calculate the interest earned by Ameesha.
→Calculate the interest earned by Ameesha.
A quadrilateral ABCD has exactly one axis of symmetry, which is not a diagonal. Show that the quadrilateral is an isosoeles trapezium.
A point P is 15 cm from the centre of a circle. The radius of the circle is 5 cm. Find the length of the tangent drawn to the circle from the point P.
If (11a² + 13b²) (11c² – 13d²) = (11a² – 13b²)(11c² + 13d²), prove that a : b :: c : d.
Ramesh chooses a date at random in january for a party.
Find the probability that the choose :
(1) a Wednesday
(2) a friday
(3) a tuesday or a saturday.
| January | |||||
| Mon | 6 | 13 | 20 | 27 | |
| Tue | 7 | 14 | 21 | 28 | |
| Wed | 1 | 8 | 15 | 22 | 29 |
| Thurs | 2 | 9 | 16 | 23 | 30 |
| Fri | 3 | 10 | 17 | 24 | 31 |
| sat | 4 | 11 | 18 | 25 | |
| Sun | 5 | 12 | 19 | 26 | |
(1) a Wednesday
(2) a friday
(3) a tuesday or a saturday.
Solve the following equation by factorization $\frac{1}{x+6}+\frac{1}{x-10}=\frac{3}{x-4}$
→