Question
Prove the following trigonometric identities.
If $\cos\text{A}+\cos^2\text{A}=1,$ prove that $\sin^2\text{A}+\sin^4\text{A}=1.$
If $\cos\text{A}+\cos^2\text{A}=1,$ prove that $\sin^2\text{A}+\sin^4\text{A}=1.$
✓
Answer
$\cos\text{A}+\cos^2\text{A}=1$
$\cos\text{A}=1-\cos^2\text{A}$
$\cos\text{A}=\sin^2\text{A}$
$\text{L.H.S}=\sin^2\text{A}+\sin^4\text{A}$
$=\sin^2\text{A}+(\sin^2\text{A})^2$
$=\sin^2\text{A}+(\cos\text{A})^2$
$=\sin^2\text{A}+\cos^2\text{A}$
$=1=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
$\cos\text{A}=1-\cos^2\text{A}$
$\cos\text{A}=\sin^2\text{A}$
$\text{L.H.S}=\sin^2\text{A}+\sin^4\text{A}$
$=\sin^2\text{A}+(\sin^2\text{A})^2$
$=\sin^2\text{A}+(\cos\text{A})^2$
$=\sin^2\text{A}+\cos^2\text{A}$
$=1=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Heights of 50 students of class X of a school are recorded and following data is obtained:
Find the median height of the students.
| Height (in cm): | 130-135 | 135-140 | 140-145 | 145-150 | 150-155 | 155-160 |
| Number of Students: | 4 | 11 | 12 | 7 | 10 | 6 |
Find the median height of the students.
For the following statments state whether true (T) or false(F):
Any two rectangles are similar.
Any two rectangles are similar.
In the given figure, $O$ is the centre of a circle. PT and PQ are tangents to the circle from an external point $P$. If $\angle T P Q=70^{\circ}$, find $\angle T R Q$.
→Find whether the following equations have real roots. If real roots exist, find them.
$-2x^2 + 3x + 2 = 0.$
→$-2x^2 + 3x + 2 = 0.$
The diameter of a sphere is 42cm. It is melted and drawn into a cylindrical wire of diameter 2.8cm. Find the length of the wire.
In the given figure, $O$ is the centre of the circle and $TP$ is the tangent to the circle from an external point $T$ . If $\angle P B T=30^{\circ}$, prove that $BA : AT =2: 1$.
→In FIg. check whether AD is the bisector of $\angle\text{A}$ of $\triangle\text{ABC}$ in the following.
AB = 4cm, AC = 6cm, BD = 1.6cm and CD = 2.4cm.
→AB = 4cm, AC = 6cm, BD = 1.6cm and CD = 2.4cm.
Solve the following equations by using the method of completing the square:
$8x^2 - 14x - 15 = 0$
→$8x^2 - 14x - 15 = 0$
As observed from the top of a light-house, $100 m$ high above sea level, the angle of depression of a ship, sailing directly towards it, changes from $30^{\circ}$ to $60^{\circ}$. Determine the distance travelled by the ship during the period of observation. $($ Use $\sqrt{3}=1.732)$
→For any positive integer n, prove that $n^3 - n$ is divisible by $6.$
→