Download App

Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities5 Marks
Question
Prove the following trigonometric identities.
$(1+\tan^2\text{A})+\Big(1+\frac{1}{\tan^2\text{A}}\Big)=\frac{1}{\sin^2\text{A}-\sin^4\text{A}}$
✓

Answer

We need to prove $(1+\tan^2\text{A})+\Big(1+\frac{1}{\tan^2\text{A}}\Big)=\frac{1}{\sin^2\text{A}-\sin^4\text{A}}$
Using the property $1+\tan^2\theta=\sec^2\theta,$ we get
$\text{L.H.S}=(1+\tan^2\text{A})+\Big(1+\frac{1}{\tan^2\text{A}}\Big)=\sec^2\text{A}+\Big(\frac{\tan^2\text{A}+1}{\tan^2\text{A}}\Big)$
$=\sec^2\text{A}+\Big(\frac{\sec^2\text{A}}{\tan^2\text{A}}\Big)$
Now, using $\sec\theta=\frac{1}{\cos\theta}\text{ and }\tan\theta=\frac{\sin\theta}{\cos\theta},$ we get
$\sec^2\text{A}+\Big(\frac{\sec^2\text{A}}{\tan^2\text{A}}\Big)=\frac{1}{\cos^2\text{A}}+\Bigg(\frac{\frac{1}{\cos^2\text{A}}}{\frac{\sin^2\text{A}}{\cos^2\text{A}}}\Bigg)$
$=\frac{1}{\cos^2\text{A}}+\Big(\frac{1}{\cos^2\text{A}}\times\frac{\cos^2\text{A}}{\sin^2\text{A}}\Big)$
$=\frac{1}{\cos^2\text{A}}+\frac{1}{\sin^2\text{A}}$
$=\frac{\sin^2\text{A}+\cos^2\text{A}}{\cos^2\text{A}(\sin^2\text{A})}$
Further, using the peroperty, $\sin^2\theta+\cos^2\theta=1,$ we get
$\frac{\sin^2\text{A}+\cos^2\text{A}}{\cos^2\text{A}(\sin^2\text{A})}=\frac{1}{\cos^2\text{A}(\sin^2\text{A})}$
$=\frac{1}{(1-\sin^2\text{A})(\sin^2\text{A})}\ (\text{using}\cos^2\theta=1-\sin^2\theta)$
$=\frac{1}{\sin^2\text{A}-\sin^4\text{A}}=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

Solve the following quadratic equation:$12abx^2 - (9a^2 - 8b^2)x - 6ab = 0$
→
A student was asked to make a model shaped like a cylinder with two cones attached to its ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its total length is 12 cm . If each cone has a height of 2 cm , find the volume of air contained in the model.
→
An iron pillar has some part in the form of a right circular cylinder and remaining in the form of a right circular cone.The radius of the base of each of cone and cylinder is $8 \ cm.$ The cylindrical part is $240 \ cm$ high and the conical part is $36 \ cm$ high. Find the weight of the pillar if one cubic $\ cm$ of iron weighs $7.8$ grams.
→
In the given figure, D is the mid-point of side BC and $\text{AE}\perp\text{BC}.$ If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that:
  1. $\text{b}^2=\text{p}^2+\text{ax}+\frac{\text{a}^2}{4}$
  2. $\text{c}^2=\text{p}^2-\text{ax}+\frac{\text{a}^2}{4}$
  3. $\text{b}^2+\text{c}^2=2\text{p}^2+\frac{\text{a}^2}{4}$
→
In Figure, from a solid cube of side 7 cm , a cylinder of radius 2.1 cm and height 7 cm is scooped out. Find the total surface area of the remaining solid.
Image
→
A solid rectangular block of dimensions 4.4m, 2.6m and 1m is cast into a hollow cylindrical pipe of internal radius 30cm and thickness ​5cm. Find the length of the pipe.
→
In an AP, the first term is -4, the last term is 29 and the sum of all its terms is 150. Find the common difference.
→
Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.
→
A quadrilateral is drawn to circumscribe a circle. Prove that the sume of opposite sides are equal.
→
Find the area and perimeter of an isosceles right-angled triangle, each of whose equal sides measures 10cm. $\big[\text{Given}:\sqrt{2}=1.41\big]$
→