Question
Prove the following trigonometric identities.
$\tan^2\text{A}+\cot^2\text{A}=\sec^2\text{A}\text{ cosec}^2\text{A}-2$
$\tan^2\text{A}+\cot^2\text{A}=\sec^2\text{A}\text{ cosec}^2\text{A}-2$
✓
Answer
$\text{L.H.S}=\tan^2\text{A}+\cot^2\text{A}$
$=\sec^2\text{A}-1+\text{cosec}^2\text{A}-1$
$=\sec^2\text{A}+\text{cosec}^2\text{A}-2$
$=\frac{1}{\cos^2\text{A}}+\frac{1}{\sin^2\text{A}}-2$
$=\frac{\sin^2\text{A}+\cos^2\text{A}}{\cos^2\text{A}\sin^2\text{A}}-2$
$=\frac{1}{\cos^2\text{A}\sin^2\text{A}}-2$
$=\sec^2\text{A}\text{ cosec}^2\text{A}-2=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
$=\sec^2\text{A}-1+\text{cosec}^2\text{A}-1$
$=\sec^2\text{A}+\text{cosec}^2\text{A}-2$
$=\frac{1}{\cos^2\text{A}}+\frac{1}{\sin^2\text{A}}-2$
$=\frac{\sin^2\text{A}+\cos^2\text{A}}{\cos^2\text{A}\sin^2\text{A}}-2$
$=\frac{1}{\cos^2\text{A}\sin^2\text{A}}-2$
$=\sec^2\text{A}\text{ cosec}^2\text{A}-2=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
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
What must be subtracted from the polynomial $f(x) = x^4 + 2x^3 − 13x^2 − 12x + 21$ so that the resulting polynomial is exactly divisible by $x^2 − 4x + 3?$
→Are the given numbers $1, 3, 9, 27, ........$ forms an $AP$? If It forms an $AP$, then find the common difference d and write the next three terms.
→Write a quadratic polynomial, sum of whose zeros is $2\sqrt{3}$ and their product is $2$.
→The area of a circular playground is $22176 m^2$. Find the cost of fencing this ground at the rate of ₹ 50 per metre.
→In the figure, altitudes AD and CE of $\triangle$ABC intersect each other at the point P. Show that: $\vartriangle AEP \sim \vartriangle CDP$
→How many three digit natural numbers are divisible by $7 ?$
→Has the rational number $\frac{441}{2^2\times5^7\times7^2}$ terminating or a non terminating decimal representation?
→Find the distance between the points:
$A(7, -4)$ and $B(-5, 1)$
→$A(7, -4)$ and $B(-5, 1)$
In an $AP,$ if $S_5+S_7=167$ and $S_{10}=235, $ then find the $AP,$ where $S_n$ denotes the sum of first terms.
→If $5\cot\theta=3,$ show that the value of $\Big(\frac{5\sin\theta-3\cos\theta}{4\sin\theta+3\cos\theta}\Big)$ is $\frac{16}{29}.$
→