Download App

Trigonometric Functions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Functions3 Marks
Question
Prove the following identities: $\sin^6\text{x}+\cos^6\text{x}=1−3\sin^2\text{x}\cos^2\text{x}$
✓

Answer

$\text{L.H.S}=\sin^{6}\text{x}+\cos^{6}\text{x}$ $=\big(\sin^{2}\text{x}\big)^{3}+\big(\cos^{6}\text{x}\big)$ $=\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)$ $\bigg[\big(\sin^{2}\text{x}\big)^{2}-\big(\sin^{2}\text{x}\big)-\sin^{2}\text{x}\cos^{2}\text{x}+\big(\cos^{2}\text{x}\big)\bigg]$ $\big(\because\text{a}^{3}+\text{b}^{3}=\big(\text{a}+\text{b}\big)\big(\text{a}^{2}-\text{ab}+\text{b}^{2}\big)$ $=\big(\sin\text{x}\big)^{2}+\big(\cos\text{x}\big)^{2}$ $\Big[\big(\sin^{2}\text{x}\big)^{2}+2\sin^{2}\text{x}\cos^{2}\text{x}-2\sin^{2}\text{x}\cos^{2}\text{x}-\sin^{2}\cos^{2}\text{x}\big)\Big]$ $=\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)^{2}-3\sin^{2}\text{x}\cos^{2}\text{x}$ $=\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)^{2}-3\sin^{2}\text{x}\cos^{2}\text{x}$ $=1^{2}-3\sin^{2}\text{x}\cos^{2}\text{x}$ $\big(\because\sin^{2}\text{x}+\cos^{2}\text{x}=1\big)$ $=1-3\sin^{2}\text{x}\cos^{2}\text{x}$ $=\text{R.H.S}$ $\text{L.H.S = R.H.S}$

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 "(Each question 3 marks)" from Trigonometric Functions→Trigonometric Functions — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the domain and range of the following real valued functions: $\text{f(x)}=\sqrt{\text{x}^2-16}$
→
The radius of a circle is 30cm. Find the length of an arc of this circle, if the length of the chord of the arc is 30cm.
→
Show that the ratio of the sum of the first n terms of a G.P. to the sum of terms from $(\text{n}+1)^\text{th}\text{ to }(2\text{n})^\text{th}\text{ terms is }\frac{1}{\text{r}^\text{n}}.$
→
Prove that: $3\sin\frac{\pi}{6}\sec\frac{\pi}{3}-4\sin\frac{5\pi}{6}\cot\frac{\pi}{4}=1$
→
Find the sum to n terms of the series $1 \times 2+2 \times 3+3 \times 4+4 \times 5+. .$
→
In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea?
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\tan\text{m}\text{x}}{\tan\text{n}\text{x}}$
→
Prove that: $\tan225^\circ\cot405^\circ+\tan765^\circ\cot675^\circ=0$
→
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from $(n + 1)^{th}$ to $(2n)^{th}$ term is $\frac { 1 } { r ^ { n } }$.
→
Convert in the polar form: $\frac{{1 + 7i}}{{{{(2 - i)}^2}}}$
→