Prove that: ( 4 +x)+ ( 4 -x)=2 2x - Vidyadip

Question

Prove that: $\tan(\frac{\pi}{4}+\text{x})+\tan(\frac{\pi}{4}-\text{x})=2\sec2\text{x}$

✓

Answer

$\text{LHS}=\tan\Big(\frac{\pi}{4}+\text{x}\Big)+\tan\frac{\pi}{4}-\text{x}\Big)$ $=\frac{\tan\frac{\pi}{4}+\tan\text{x}}{1-\tan\frac{\pi}{4}\tan\text{x}}+\frac{\tan\frac{\pi}{4}-\tan\text{x}}{1+\tan\frac{\pi}{4}\tan\text{x}}$ $=\frac{1+\tan\text{x}}{1-\tan\text{x}}+\frac{1-\tan\text{x}}{1+\tan\text{x}}$ $\Big[\because\tan\frac{\pi}{4}=1\Big]$ $=\frac{(1+\tan^2\text{x}+2\text{x})+(1+\tan^2\text{x}-1\tan\text{x})}{(1-\tan\text{x})(1+\tan\text{x})}$ $=\frac{1(1+\tan^2\text{x})}{1-\tan^2\text{x}}$ $=\frac{1\sec^2\text{x}}{1-\frac{\sin^2\text{x}}{\cos^2\text{x}}}$ $[\because\sec^2\text{x}=1+\tan^2\text{x}]$ $=\frac{2\sec^2\text{x}.\cos^2\text{x}}{\cos^2\text{x}-\sin62\text{x}}$ $[\because\sec-\frac{1}{\cos\text{x}}]$ $=\frac{2}{\cos2\text{x}}$ $=2\sec2\text{x}=\text{RHS}$

Need a full question paper?

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

Start Generating Free

Explore more

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

Similar questions

A letter is chosen at random from the word ASSASSINATION find the probability that letter is a vowel.

Find the sum of the following series to n terms: $2^2 + 4^2 + 6^2 + 8^2 + ....$

For all positive integers n, show that ${^{2\text{n}}}\text{C}_{\text{n}}+{^{2\text{n}}}\text{C}_{\text{n}-1}=\frac{1}{2}\big({^{2\text{n+2}}}\text{C}_{\text{n}+1}\big).$

Let A = {x : x $\in$ N}, B = {x : x = 2n, n $\in$ N}, C = {x : x = 2n - 1, n $\in$ N} and D = {x : x is a prime natural number}. Find:
$\text{A}\cap\text{B}$

If a, b, c, d are in G.P., prove that $(a^n + b^n), (b^n + c^n), (c^n + d^n)$ are in G.P.

If P(n ,5) = 20.P(n, 3), find n.

Find the multiplicative inverse of the following complex numbers: $1-\text{i}$

Let A = {1, 2, 3, 4, 5, 6}. Let R be a relation on A defined by
{(a, b) : a, b ∈ A, b is exactly divisible by a}

Writer R in roster form.
Find the domain of R.
Find the range of R.

Prove that: $\tan13\text{x}-\tan9\text{x}-\tan4\text{x}=\tan13\text{x}\tan9\text{x}\tan4\text{x}$

A survey of 500 television viewers produced the following information; 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and football, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three game?