Question
Prove the following trigonometric identities.
$\tan^2\theta-\sin^2\theta=\tan^2\theta\sin^2\theta$
$\tan^2\theta-\sin^2\theta=\tan^2\theta\sin^2\theta$
✓
Answer
$\text{L.H.S}=\tan^2\theta-\sin^2\theta$
$=\frac{\sin^2\theta}{\cos^2\theta}-\frac{\sin^2\theta}{1}$
$=\sin^2\theta\Big[\frac{1}{\cos^2\theta}-\frac{1}{1}\Big]$
$=\sin^2\theta\Big(\frac{1-\cos^2\theta}{\cos^2\theta}\Big)=\sin^2\theta\cdot\frac{\sin^2\theta}{\cos^2\theta}$
$\sin^2\theta\cdot\tan^2\theta=\text{R.H.S}$
$=\frac{\sin^2\theta}{\cos^2\theta}-\frac{\sin^2\theta}{1}$
$=\sin^2\theta\Big[\frac{1}{\cos^2\theta}-\frac{1}{1}\Big]$
$=\sin^2\theta\Big(\frac{1-\cos^2\theta}{\cos^2\theta}\Big)=\sin^2\theta\cdot\frac{\sin^2\theta}{\cos^2\theta}$
$\sin^2\theta\cdot\tan^2\theta=\text{R.H.S}$
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 is the area of the triangle formed by the points O(0, 0), A(6, 0) and B(0, 4)?
If two coins are tossed, find the probability of the following events.
Getting at least one head.
Getting at least one head.
Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.
In Δ ABC, seg BD bisects ∠ ABC. If AB = x, BC = x + 5, AD = x – 2, DC = x + 2, then find the value of x.
The circles which are not congruent touch externally. The sum of their areas is $130 \pi cm ^2$ and distance between their centres is 14 cm. Find the radii of the two circles.
→Triangle ABC and DEF are similar.If AC = 19cm and DF = 8cm, find the ratio of the area of two triangles.
The faces of a red cube and a yellow cube are numbered from 1 to 6. Both cubes are rolled. What is the probability that the top face of each cube will have the same number?
Two tangents TP and TQ are drawn to a circle with centre $O$ from an external point T. Prove that $\angle PTQ =2 \angle OPQ$.
→Determine the nature of roots of the following quadratic equation from their discriminant:
$2 x^2-3 x-4=0$
→$2 x^2-3 x-4=0$
How many terms are there in the A.P. 187, 194, 201, ..., 439 ?