Question
Prove. $(1 + \tan A \tan B)^2 + ( \tan A - \tan B)^2 = \sec ^2A \sec ^2B$
✓
Answer
$\text{LHS} =(1 + \tan A \ \tan B)^2 + ( \tan A - \tan B)^2$
$= 1 + \ \tan ^2A \ \tan ^2B + 2 \tan A \ \tan B+\ \tan ^2A + \ \tan ^2B - 2 \tan A \ \tan B$
$= 1 + \ \tan ^2A + \ \tan ^2B + \ \tan ^2A \ \tan ^2B$
$= \sec^2A + \ \tan ^2B (1 + \ \tan ^2A)$
$= \sec^2A + \ \tan ^2B \ sec^2A$
$= \sec^2A (1 + \ \tan ^2B)$
$= \sec^2A sec^2B = \text{RHS}$
$= 1 + \ \tan ^2A \ \tan ^2B + 2 \tan A \ \tan B+\ \tan ^2A + \ \tan ^2B - 2 \tan A \ \tan B$
$= 1 + \ \tan ^2A + \ \tan ^2B + \ \tan ^2A \ \tan ^2B$
$= \sec^2A + \ \tan ^2B (1 + \ \tan ^2A)$
$= \sec^2A + \ \tan ^2B \ sec^2A$
$= \sec^2A (1 + \ \tan ^2B)$
$= \sec^2A sec^2B = \text{RHS}$
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
Calculate the percentage income in the following investment:
Rs 12,375 in a Rs 75 share paying 4% and available at a discount of Rs 20.
Rs 12,375 in a Rs 75 share paying 4% and available at a discount of Rs 20.
The curved surface area of a cone is 12320 cm2. If the radius of its base is 56 cm, find its height.
A circle with diameter $20 \ cm$ is drawn somewhere on a rectangular piece of paper with length $40 \ cm$ and width $30 \ cm.$ This paper is kept horizontal on table top and a die, very small in size, is dropped on the rectangular paper without seeing towards it. If the die falls and lands on paper only, find the probability that it will fall and land: outside the circle
→$\triangle \mathrm{ABC}$ is reduced by a scale factor 0.72 . If the area of $\triangle \mathrm{ABC}$ is $62.5 \mathrm{~cm}^2$, find the area of the image.
→Find the area of a circular field that has a circumference of $396$ m.
→In the figure, $PM$ is a tangent to the circle and $PA = AM$. Prove that:
(i) Δ PMB is isosceles
(ii) $PA \times PB = MB^2$
→(i) Δ PMB is isosceles
(ii) $PA \times PB = MB^2$
From the given figure, find:
(i) the co-ordinates of $A, B$ and $C$.
(ii) the equation of the line through A and parallel to $BC$.
→(i) the co-ordinates of $A, B$ and $C$.
(ii) the equation of the line through A and parallel to $BC$.
In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30°, find: angle APB
Find the sum : $243+324+432+\ldots$ up to $n$ terms.
→In Δ ABC, D and E are points on the sides AB and AC respectively. If AD= 4cm, DB=4.Scm, AE=6.4cm and EC=7.2cm, find if DE is parallel to BC or not.