Download App

Trigonometric Ratios of Multiple and Submultiple Angles — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Ratios of Multiple and Submultiple Angles4 Marks
Question
Prove that: $\frac{1}{\sin\text{(x}-\text{b})\sin\text{(x}-\text{b)}}=\frac{\cot\text{(x}-\text{a)}+\tan\text{(x}-\text{b)}}{\cos\text{(a}-\text{b)}}$
✓

Answer

$\text{L.H.S}=\frac{1}{\sin\text{(x}-\text{a)}\cos\text{(x}-\text{b)}}$ $=\frac{1}{\cos\text{(a}-\text{b)}}\Big[\frac{\sin\text{(a}-\text{b)}}{\sin\text{(x}-\text{b})\cos\text{(x}-\text{b)}}\Big]$ $=\frac{1}{\cos\text{(a}-\text{b)}}\Big[\frac{\cos\text{(x}-\text{b)-}(\text{x}-\text{b)}}{\sin\text{(x}-\text{a})\cos\text{(x}-\text{b)}}\Big]$ $=\frac{1}{\cos\text{(a}-\text{b)}}\Big[\frac{\cos\text{(x}-\text{b)cos}(\text{x}-\text{a)}+\sin\text{(x}-\text{b})\sin\text{(x}-\text{a)}}{\sin\text{(x}-\text{a})\sin\text{(x}-\text{b)}}\Big]$ $=\frac{1}{\cos\text{(a}-\text{b})}\Big[\frac{\cos\text{(x}-\text{b)}\cos\text{(x}-\text{a)}}{\sin\text{(x}-\text{b)}\cos\text{(x}-\text{b})}+\frac{\sin\text{(x}-\text{b)}\sin\text{(x}-\text{a)}}{\sin\text{(x}-\text{b)}\cos\text{(x}-\text{b})}\Big]$ $=\frac{\cot\text{(a}-\text{b)}+\tan\text{(x}-\text{b)}}{\cos\text{(a}-\text{b)}}$ $=\frac{1}{\sin\text{(a}-\text{b})}\Big[\cot\text{(x}-\text{b)}-\cot\text{(x}-\text{b)}\Big]$ $=\frac{\cot\text{(x}-\text{a)-}\cot\text{(x}-\text{b)}}{\sin\text{(a}-\text{b)}}$ $=\text{R.H.S}$ $\therefore\ \text{L.H.S}=\text{R.H.S}$ Hence 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 "(Each question 4 marks)" from Trigonometric Ratios of Multiple and Submultiple Angles→Trigonometric Ratios of Multiple and Submultiple Angles — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove the following identities $:\frac{\sin^3+\cos^3\text{x}}{\sin\text{x}+\cos\text{x}}+\frac{\sin^3\text{x}-\cos^3\text{x}}{\sin\text{x}-\cos\text{x}}=2$
→
Prove the following by using the principle of mathematical induction for all n ∈ N:$\text{x}^{2\text{n}}-\text{y}^{2\text{n}}$ is divisible by $x + y$.
→
Evaluate the following limit: $\lim\limits_{\text{n}\rightarrow\infty}\frac{{(\text{n}+2)!}+{(\text{n}+1)!}}{{(\text{n}+2)!}+{(\text{n}+1)!}}$
→
Prove that the number of subsets of a set containing $\mathbf{n}$ distinct elements is $2^n$ for all $\mathbf{n} \in \mathbf{N}$.
→
Verify that the area of the triangle with verticies (2, 3), (-3, -1) remains invariant under the translation of axes when the origin is shifted to the point (-1, 3).
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\sqrt{6}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{3}+\sqrt{2}\big)}{\text{x}^2-6}$
→
In each of the followin find the equation of the hyperbola satisying the given conditions:

vertices $(0, \pm3)$, foci $(0, \pm5)$ [NCERT]

→
Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of -6 - 24i.
→
Solve the following equation: $\sqrt{3}\text{x}^2-\sqrt{2}\text{x}+3\sqrt{3}=0.$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}-3}{\sqrt{\text{x}-2}-\sqrt{4-\text{x}}}$
→