Prove the following identities: ^2 (1- ) + ^2 ( - ) =(1+ ) - Vidyadip

Question

Prove the following identities:
$\frac{\cos^2\theta}{(1-\tan\theta)}+\frac{\sin^2\theta}{(\sin\theta-\sin\theta)}=(1+\sin\theta\cos\theta)$

✓

Answer

$\text{L.H.S.}=\frac{\cos^2\theta}{(1-\tan\theta)}+\frac{\sin^2\theta}{(\sin\theta-\sin\theta)}$ $=\frac{\cos^2\theta}{\Big(1-\frac{\sin\theta}{\cos\theta}\Big)}+\frac{\sin^3\theta}{(\sin\theta-\cos\theta)}$ $=\frac{\cos^3\theta}{(\cos\theta-\sin\theta)}+\frac{\sin^3\theta}{(\sin\theta-\cos\theta)}$$=\frac{\cos^3\theta}{(\cos\theta-\sin\theta)}-\frac{\sin^3\theta}{\cos\theta-\sin\theta}$
$=\frac{\cos^3\theta-\sin^3\theta}{(\cos\theta-\sin\theta)}$ $=\frac{(\cos\theta-\sin\theta)\big(\cos^2\theta+\cos\theta\sin\theta+\sin^2\theta\big)}{(\cos\theta-\sin\theta)}$ $\Big[\because\text{a}^3-\text{b}^3=(\text{a}-\text{b})\big(\text{a}^2+\text{ab}+\text{b}^2\big)\Big]$ $=(1+\cos\theta\sin\theta)$ $=\text{R.H.S.}$ $\therefore\text{R.H.S.}=\text{L.H.S.}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

The inner and outer radii of a hollow cylinder are $15\ cm$ and $20\ cm$, respectively. The cylinder is melted and recast into a solid cylinder of the same height. Find the radius of the base of new cylinder.

Draw an ogive by less than method for the following data:

No. of rooms	1	2	3	4	5	6	7	8	9	10
No. of houses	4	9	22	28	24	12	78	6	5	2

If $(x + a) $is a factor of $2x^2 + 2ax + 5x + 10,$ find a.

A person, rowing at the rate of 5km/hr in still water, takes thrice as much time in going 40km upstream as in going 40km downstream. Find the speed of the stream.

In the following, determine whether the given values are solution of the given equation or not:
$x^2 + x + 1 = 0, x = 0, x = 1$

Solve the following quadratic equation:$\text{10x}-\frac{1}{\text{x}}=3$

A play ground has the shape of a rectangle, with two semi-circles on its smaller sides as diameters, added to its outside. If the sides of the rectangle are 36m and $24.5\ m$, find the area of the playground. $\Big(\text{Take }\pi=\frac{22}{7}\Big).$

Find the value of k for which the following system of equations has no solution:

Find the coordinate of the point which divides the join of A(-1, 7) and B(4, -3) in the ratio 2 : 3.

Find the coordinates of the point which divides the line segment joining $(-1, 3)$ and $(4, -7)$ internally in the ratio $3 : 4.$