Trigonometric Ratio and Identites

If L = sin2\(\left( {{\pi \over {16}}} \right)\) - sin2\(\left( {{\pi \over {8}}} \right)\) and
M = cos2\(\left( {{\pi \over {16}}} \right)\) - sin2\(\left( {{\pi \over {8}}} \right)\), then :

If the equation cos4 \(\theta\) + sin4 \(\theta\) +
\(\lambda\) = 0 has real
solutions for
\(\theta\), then
\(\lambda\) lies in the interval :

For any \(\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)\), the expression
\(3{(\cos \theta - \sin \theta )^4}\)\(+ 6{(\sin \theta + \cos \theta )^2} + 4{\sin ^6}\theta\)
equals :

The value of cos210° – cos10°cos50° + cos250° is

The value of sin 10º sin30º sin50º sin70º is :-

If cos(\(\alpha\) + \(\beta\)) = 3/5 ,sin ( \(\alpha\) - \(\beta\)) = 5/13 and
0 < \(\alpha , \beta\) < \(\pi \over 4\), then tan(2\(\alpha\)) is equal to :

The maximum value of 3cos\(\theta\) + 5sin \(\left( {\theta - {\pi \over 6}} \right)\) for any real value of \(\theta\) is :

The equation y = sinx sin (x + 2) – sin2
(x + 1) represents a straight line lying in :

The value of \(\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}}\) is -

If \(5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9\),
then the value of \(\cos 4x\) is :

If  m and M are the minimum and the maximum values of
4 + \({1 \over 2}\) sin2 2x \(-\) 2cos4 x, x \(\in\) R, then M \(-\) m is equal to :

Let \(f_k\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\) where \(x \in R\) and \(k \ge \,1.\)
Then \({f_4}\left( x \right) - {f_6}\left( x \right)\,\,\) equals :

The expression \({{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}\) can be written as:

If \(A = {\sin ^2}x + {\cos ^4}x,\) then for all real \(x\):

Let \(\cos \left( {\alpha + \beta } \right) = {4 \over 5}\) and \(\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},\) where \(0 \le \alpha ,\,\beta \le {\pi \over 4}.\)
Then \(tan\,2\alpha\) =

Let A and B denote the statements
A: \(\cos \alpha + \cos \beta + \cos \gamma = 0\)
B: \(\sin \alpha + \sin \beta + \sin \gamma = 0\)
If $$\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left(...

If \(0 < x < \pi\) and \(\cos x + \sin x = {1 \over 2},\) then \(\tan x\) is :

If \(u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}\theta }\)
then the difference between the maximum and minimum values of \({u^2}\) is given by :

Let \(\alpha ,\,\beta\) be such that \(\pi < \alpha - \beta < 3\pi\).
If \(sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}\) and \(\cos \alpha + \cos \beta = - {{27} \over {65}}\) then the value of $$\cos {{\alpha - ...