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

Question

Prove the following identities:
$(1+\tan\alpha\tan\beta)^2+(\tan\alpha-\tan\beta)^2=\sec^2\alpha\sec^2\beta$

✓

Answer

$\text{L.H.S}=(1+\tan\alpha\tan\beta)^2+(\tan\alpha-\tan\beta)^2$
$=1(\tan\alpha+\tan\beta)^2+2.1\tan\alpha\tan\beta\\\ \ +(\tan\alpha)^2+(\tan\beta)^2-1\tan\alpha.\tan\beta$ $(\text{Using(a+b)}^2=\text{a}^2+\text{b}^2+2\text{ab and (a-b)}^2=\text{a}^2+\text{b}^2-2\text{ab})$
$=1+\tan^2\alpha+\tan^2\beta+2\tan\alpha\tan\beta+\tan^2\alpha+\tan^2\beta-2\tan\alpha\tan\beta$
$=1+\tan^2\alpha+\tan^2\alpha+\tan^2\beta+\tan^2\beta$
$=\sec^2\alpha+\tan^2\beta(1+\tan^2\alpha)$$(\because1+\tan^2\alpha=\sec^2\alpha)$
$=\sec^2\alpha+\tan^2\beta.\sec^2\alpha$
$=\sec^2\alpha(1+\tan^2\beta)$
$=\sec^2\alpha.\sec^2\beta$ $(\because1+\tan^2\beta=\sec^2\beta)$
$=\text{R.H.S}$
$\text{Proved}$

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 "Solve the Following Question.(4 Marks)" from Trigonometric Functions→Trigonometric Functions — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the square roots of
6 + 8i

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{2}+1\big)}{\text{x}^2-2}$

Prove that
$\cos\frac{\pi}{65}\cos\frac{2\pi}{65}\cos\frac{4\pi}{65}\cos\frac{8\pi}{65}\cos\frac{16\pi}{65}\cos\frac{32\pi}{65}=\frac{1}{64}$

If a, b, c, are in G.P., prove that the following are also in G.P.
$\text{a}^2,\text{b}^2,\text{c}^2$

If the length of the perpendicular from the point (1, 1) to the line ax - by + c = 0 be unity, show that $\frac{1}{\text{c}}+\frac{1}{\text{a}}-\frac{1}{\text{b}}=\frac{\text{c}}{2\text{ab}}.$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\bigg\{\frac{1}{\text{x}-2}-\frac{2(2\text{x}-3)}{\text{x}^2-3\text{x}^2+2\text{x}}\bigg\}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{5\text{x}-4}-\sqrt{\text{x}}}{\text{x}^2-1}$

Find the matrices $A$ and $B$, if $2 A-B=\left[\begin{array}{ccc}6 & -6 & 0 \\ -4 & 2 & 1\end{array}\right]$ and $A-2 B=\left[\begin{array}{ccc}3 & 2 & 8 \\ -2 & 1 & -7\end{array}\right]$

Calculate the mean deviation of the following income groups of five and seven members from their medians:

$I$ Income in $₹$	$II$ Income in $₹$
$4000$	$3800$
$4200$	$4000$
$4400$	$4200$
$4600$	$4400$
$4800$	$4600$
	$4800$
	$5800$

If $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 3 & 1 & 2\end{array}\right], B=\left[\begin{array}{lll}2 & 1 & -4 \\ 3 & 5 & -2\end{array}\right]$ and $C=\left[\begin{array}{ccc}0 & 2 & 3 \\ -1 & -1 & 0\end{array}\right]$, verify that $(A+2 B+$

$3 C)^{\top}=A^{\top}+2 B^{\top}+3 C^{\top}$.